






■•<H'. .'"/r 



rm 



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IN MEMORIAM 
FLORIAN CAJORI 





"SCOTT'S 2IEJi- 



BRADY'S 

BAIiTIMORE. Ml 



^/U:^^-m.^..m>' y/^ ^"^^^ 



INTRODUOTIOiV 



TO 






WITH 

NOTES AND OBSERVATi:^:^^ 




USE OF SCHOOLS AND PLACES OF 



DESIGNED FOR THt j^ / 



ON THE 

APPLICATION OF ALGEBRA TO GEOMETRY. 



By JOHN BONNYCASTLE, 

Professor of Mathematics in the Royal Military Academy, Woolwich. 



SECOND NEW-YORK, FROM THE ELEV£NTH LONDON EDITION". 



SIVISED, COHKECTiD, AND ENLARGED, WITH A VARIETY OF EXAMPLSfj 
AND MANY OTHER DSEFITL ADDITIONS, 

By JAIVIES RYAN, 
Private Teacher of Mathematics. 



■ ■■ Infennas didicisse fideliter arte! 
Kmolltt mores, nee sinit esse feres. Ovi^^ 

NEW-YORK : 
PUBLIStiEB BY EVERT DUYCKINCK AND GEORGE LONG, 

1822. 



QA 



i-> 



<r 1 



» ^ 



U2. 



CAJORl 



Stfu^o n Datrict of Ifew-Tork, M. 

BE IT REMEMBERED, That on the twenty-eighth day of December, in the 
forty-sixth year of the Independence of the United StPtes of America, Oeorge Long, 
of the said district, hath deposited in this office the title of a book, the right whereof 
ke claims as proprietor/ in the words following, to wit ; 

" An Introduction to Algebra, with Notes and Observations; designed for the Use 
of Schools and places of Public Education. To which is added an Appendix, on 
the Application of Algebra to Geometry. By John Bonnycastle, Professor of Ma- 
thematics in the Royal Mililary Academy, Woolwich. Second New-Tork from the 
£leTenth London Edition. Revised, corrected, and enlarged, with a variety of Ex- 
amples, and many other useful Additions, by James Ryan, Private Teacher of Ma- 
tbeisaticf. 

Ingenuas didicisse fideliter artes 

EmoUit mores, ne« sinit esse feros. Ovid." 

In conformity to the act of Congress of the United States, entitled, « An a«t 
for the encouragement of learning, by securing the copies of maps, charts, and books, 
to the authors and proprietors of such copies, during the time therein mentioned." 
And also to an act, entitled, " An act, supplementary to an act, entitled, An act for the 
encouragement of learning, by securing the copies of maps, charts, and Jjooks, to the 
authors aiu) proprietors of such copies, during the times therein mentioned, and ex- 
tending tte benefits thtreof to the arts of designing, engraring, and etching historical 
{Old other prints." 

JAMES WLL^ 

CUrk of th» Southern Diatricl of Iftw-Yark, 



PREFACE. 



THE powers of the mind, like th<M(j|of the body are in- 
creased by frequent exertion ; application and industry 
supply the place of genius and invention ; and even the 
creative faculty itself may be sftengthened and improved 
by use and perseverance. Uncultivated nature is uniform- 
ly rude and imbecile, it being by imitation alone that we 
at first acquire knowledge, and the means of extending its 
bounds. A just and perfect acquaintance with the simple 
elements of science, is a necessary step towards our future 
progress and advancement ; and this, assisted by laborious 
investigation and habitual inquiry, will constantly lead to 
eminence and perfection. 

Books of rudiments, therefore, concisely written, well 
digested, and methodically arranged, are treasures of ines- 
timable value ; and too many attempts cannot be made to 
render them perfect and complete. When the first prin- 
ciples of any art or science are firmly fixed and rooted in 
the mind, their application soon becomes easy, pleasant, 
and obvious ; the understanding is delighted and enlarged ; 
we conceive clearly, reason distmctly, and form just and 
and satisfactory conclusions. But. on the contrary, whea 
the mind, instead of reposing on the stability of truth and 
received principles, is wandering in doubt and uncertain- 
ty, our ideas will necessarily be confused and obscure ; and 
every step we take must be attended with fresh difficulties 
and endless perplexity. 



m^ -f« ^€ic 



iv PREFACE. 

That the grounds, or fundamental parts, of every sci- 
ence, are dull and unentertaining, is a complaint univer- 
sally made, anfl a truth not to be denied ; but then, what 
is obtained with difficulty is usually remembered with ease ; 
and what is purchased wth pain is often possessed with 
pleasure. The seeds of knowledge are sown in every soil, 
but it is by proper culture alone that they are cherished 
and brought to maturity. A few years of early and assi- 
duous application, never fails to procure us the reward of 
our industry ; and who is there, who knows the pleasures 
and advantages which the sciences afford, that would think 
bis time, in this case mis-spent, or his labours useless ? 
Riches and honours are the gifts of fortune, casually be- 
stowed, or hereditarily received, and are frequently abus- 
ed by their possesgirs ; but the superiority of wisdom and 
knowledge 1;= a preeminence of merit, which originates 
with the man, and is the noblest of all distinctions. 

Nature, bountiful and wise in all things, has provided us 
with an infinite variety of scenes, both for our instruction 
and entertainment ; and, like a kind and indulgent parent, 
admits all her children to an equal participation of her 
blessings. But, as the modes, situations and circumstances 
of life are various, so accident, habit and education, have 
each their predominating influence, and give to every mind 
its particular bias. Where examples of excellence are 
wanting, the attempts to attain it are but few ; but emi- 
nence excites attention, and produces imitation. I'o raise 
the curiosity, and to awaken the listless and dormant pow- 
ers of younger minds, we have only to point out to them a 
valuable acquisition, and the means of obtaining it ; the 
active principles are immediately put into motion, and the 
certainty of the conquest is ensured from a determination 
to conquer. 

But, of all the sciences which serve to call forth this 
spirit of enterprise and inauiry. there are none more emi- 
nently useful than Mathematics. By an early attachment 
to these elegant and sublime studies, we acquire a habit of 
reasoning, and an elevation of thought, which fixes the 
mind, and prepares it for every other pursuit. From a 
few simple axioms, and evident principles, we proceed 



PREFACE. V 

gradually to the most general propositions, and remote an- 
alogies : deducing one truth from another, in a chain oi* 
argument well connected and logically pursued ; which 
brings us at last, in the most satisfactory manner, to the 
conclusion, and serves as a general direction in all our in- 
quiries after truth. 

And it is not only in this respect that mathematical learn- 
ing is so highly valuable ; it is, likewise equally estimable 
for its practical utility. Almost all the works of art and 
devices of man, have a dependence upon its principles, and 
are indebted to it for their origin and perfection. The 
cultivation of these admirable sciences is, therefore, a thing 
of the utmost importance, and ought to be considered as a 
principal part of every liberal and well-regulated plan of 
education. They are the guide of our youth, the perfec- 
tion of our reason, and the foundation of every great and 
noble undertaking. 

From these considerations, I have been induced to com- 
pose an introductory course of mathematical science ; and- 
from the kind encouragement which I have hitherto re- 
ceived, am not without hopes of a continuance of the same^ 
candour and approbation. Considerable practice as a 
teacher, and a long attention to the difficulties and obstruc- 
tions which retard the progress of learners in general, have 
enabled me to accommodate myself the more easily to their 
<^apacities and understandings. And as an earnest desire 
of promoting and diffusing useful knowledge is the chief 
motive for this undertaking, so no pains or attention shall 
be wanting to make it as complete and perfect as possible. 

The subject of the present performance is Algebra ; 
which is one of the most important and useful branches oi' 
those sciences, and may be justly considered as the key to 
all the rest. Geometry delights us by the simplicity of its 
principles, and the elegance of its demonstrations ; Arith- 
metic is confined in its object, and partial in its application ; 
but Algebra, or the Analytic Art, is general and comprehen= 
sive, and may be applied with success in all cases where 
truth is to be obtained an'^' proper data can be established. 

To trace this science to its birth, and to point out the 
various alterations and improvements it has undergone in 

A g 



vi PREFACE. 

its progress, would far exceed the lioiits of a preface.* It 
will be sufficient to observe that the invention is of the 
highest antiquity, and has challenged the praise and admi- 
ration of all ages. Diophanius, a Greek mathematician, of 
Alexandria in Egypt, who flourished in or about the third 
century after Christ, appears to have been the first, among 
the ancients, who applied it to the solution of indetermi- 
nate or unlimited problems ; but it is t» the moderns that 
we are principally indebted for the most curious refine- 
ments of the art, and its great and extensive usefulness in 
every abstruse and difficult inquiry. Ne-wton, Maclaurin^ 
Sanderson, Simpson, and Emerson, among our own country- 
men, and Clairant, Euler, Lagrange and Lacroix, on the 
continent, are those who have particularly excelled in this 
respect ; and it is to their works that 1 would refer the 
young siudent, as the patterns of elegance and perfection. 
The following compendium is formed entirely upon the 
model of those writers, and is intended as a useful and ne- 
cessary introduction to them. Almost every subject, which 
belongs to pure Algebra, is concisely and distinctly ti;;eated 
of; and no pains have been spared to make the whole as 
easy and intelligible as possible. A great number of ele- 
mentary books have already been written upon this sub- 
ject ; but there are none, which I have yet seen, but what 
appear to me to be extremely defective. Besides being 
totally unfit for the purpose of teaching, they are gene- 
rally calculated to vitiate the taste, and mislead the judg- 
ment. A tedious and inelegant method prevails through 
the whole, so that the beauty of the science is generally 
destroyed by the clumsy and awkward manner in which it 
is .treated; and the learner, when he is afterwards intro- 
duced to some of our best writers, is obliged, in a great 
measure, to unlearn and forget every thing which he has 
been at so much pains in acquiring. 

There is a certain taste and elegance in the sciences, 91 



* Those -who are desirous of a knowledge of this kind, may consult the In- 
iioduction to my Treatise on Algebra; where they will find a regular histo- 
rical detail of tlie rise and progress of the science, from its first rude begin- 
nings to the present times. 



PREFACE. Tii 

well as in every branch of polite literature, which is only 
to be obtained from the best authors, and a judicious use 
of their instructions. To direct the student in his choice 
of books, and to prepare him properly for the advantages 
he may receive from them, is therefore, the business of 
every writer who engages in the humble, but useful task 
of a preliminary tutor. This information I have been 
careful to give, in every part of the present performance, 
where it appeared to be in the least necessary ; and, 
though the nature and confined limits of my plan admitted 
not of diffuse observations, or a formal enumeration of par- 
ticulars, it is presumed nothing of real use and importance 
has been omitted. My principal object was to consult the 
ease, satisfaction, and accommodation of the learner ; and 
the favourable reception the work has met with from the 
public, has afiforded me the gratification of believing that 
my labours have not been unsuccessfully emyloyed. 



ADVERTISEMENT. 



The present performance having passed 
through a number of editions since the time 
of its first publication, without any material 
alterations having been made, either with re- 
spect to its original plan, or the manner in 
which it was executed, 1 have been induced, 
from the flattering approbation it has constant- 
ly received, to undertake an entire revision of 
the work ; and, by availing myself of the im- 
provements that have been subsequently made 
in the science, to render it still more deserv- 
ing the public favour. 

In its present state, it may be considered as 
a copious abridgment of the most practical 
and useful parts of my larger work, entitled, 
A Treatise on Algebra, in 2 vols. 8vo. published 
in 1813 ,• from which, except in certain cases 
where a different mode of proceeding appear- 
ed to be necessary, it has been chiefly com- 
piled : great care having been taken, at the 



same lime, to adapt it, as much as possible, to 
the wants of learners, and the general pur- 
poses of instruction, agreeably to the design 
with which it was first written. 

With this view, as well as in compliance 
with the wishes of several intelligent teachers, 
I have also been led to subjoin to li^ by way of 
an Appendix, a small tract on the application 
of Algebra to the solution of Geometrical 
Problems; which, it is hoped, will prove ac- 
ceptable to such classes of students as may 
not have an opportunity of consulting more 
voluminous and expensive works on this inte- 
resting branch of the science. 

John Bonnycastle. 



RoYAi, Military Academy, 
Woolwich. 
Oc4ober22, 1815, 



ADVERTISEMENT 

TO 

THE SECOND NEW-YORK EDITION. 



It would be superfluous to advance any thing 
2n commendation of "Bonnycastle's Introduc- 
lion to Algebra," as the number of European 
editions, and the increase of demand for it 
since Its publication in this country, are suffi- 
cient proofs of its great utility. 

But to make it universally useful both to 
the tutor and scholar, I have given in this edi^ 
lion, the answers that were omitted by the au- 
thorin the original. 

In the course of the work, particularly in 
Addition, Subtraction, Multiplication, Divi- 
sion, Fractions, Simple Equations, and Quad- 
ratics, I have added a great variety of prac- 
tical examples, as being essentially necessary 
to exercise young students in tiife. elementary 
principles. ~ "^ 

Several new rules are introduced, those of 
principal note are the following: Case 12. 



XI 



Surds, containing two rules for finding any 
root of a Binomial Surd, the Solution of Cu- 
bics by Converging Series, the Solution of Bi- 
quadratics by Simpson's and Euler's methods: 
all these rules are investigated in the plainest 
manner possible, with notes and remarks, in- 
terspersed throughout the work, containing 
some very useful matter. 

There is also given all the Dipohantine Ana- 
lysis, contained in Bonnycastle's Algebra, Vol. 
J. 8vo. 1820., being a methodical abstract of 
this part of the science, which comprehends 
most of the methods hitherto known for resolv- 
ing problems of this kind, and will be found a 
ready compendium for such readers as may ac- 
quire some knowledge of the Analytic Art. 

JAMES RYAN, 

JVew-York, Jan. 1, 1822. 



P. S. A new and correct Key to the present edition, com> 
prising in an easy and elegant manner, the solutions of the 
questions, is now ready for the press, and will be speedily 
published. /. R. 



L 



CONTENTS. 



DSFI5IT10R3 1 

Q 

Addition ...-•••• 

12 
Subtractwn • 

Multiplication 

Division * 

Algebraic Fractions *'' 

Involution, or the Raising of Powers 44 

Evolution, or the Extraction of Roots • 47 

Of Irrational Quantities, or Surds 54 

Of Arithmetical Proportion and Progression 87 

OfGeometrical Proportion and Progression 91 

Of Equations ^ 

Of the Resolution of Simple Equations 101 

Miscellaneous Questions 119 

Of Quadratic Equations 128 

Questions producing Quadratic Equations 139 

Of Cubic Equations . 148 

Of the Solution of Cubic Equations '. • 150 

Of the Solution of Cubic Equations by Converging Series 156 

Of the Resolution of Biquadratic Equations 168 

To find the Roots of Equations by Approximation 176 

To find the Roots of Exponential Equations 182 

Of the Binomial Theorem 184 

Of the Indeterminate Analysis 190 

Of the Diophantine Analysis 201 

Of the Summation and Interpolation of Series 233 

Of Logarithms 256 

Multiplication by Logarithms 268 

Division by Logarithms . . . 271 

The Rule of Three by Logarithms 273 

Involution, by Logarithms 275 

Evolution, by Logarithms 277 

A Collection of Miscellaneous Questions 280 

Appendix, on the Application of Algebra to Geometry 286 



ALGEBIIA. 

Algebra is the science which treats of a general me- 
thod of performing calculations, and resolving mathemati- 
cal problems, by means of the letters of the alphabet. 

Its leading rules are the same as those of arithmetic ; 
and the operations to be performed are denoted by the 
following characters : 

-\- plus, or more, the sign of addition ; signifying that 
the quantities between which it is placed are to be added 
together. 

Thus, a-\-b shows that the number, or quantity, repre- 
sented by b, is to be added to that represented by a ; and 
is read a plus b. 

— minus, or less, the sign of subtraction ; signifying 
that the latter of the two quantities between which it is 
placed is to be taken from the former. 

Thusa— fe shows that the quantity represented by 5 
is to be taken from that represented by a ; and is read a 
minus b. 

Also, a>rb represents the difference of the two quan- 
tities a and b, when it is not known wliich of them is the 
greater. 

X info, the sign of multiphcation ; signifying that the 
quantities between which it is placed are to be multiplied 
together. 

Thus, aXb shows that the quantity represented by a 
is to be multiplied by that represented by b ; and is read 
a into b. 

The multiplication of simple quantities is also frequent- 
ly denoted by a point, or by joining the letters together in 
the form of a word. 



2 DEFINITIONS. 

Thus, aXh, 4". hi and ah, all signify the product of a 
and b : also, 3 xa, or 3a, vs the product of 3 and a ; and is 
rea(J;3;tipies a"/ ,..•,:: 

-f-" bij, the sign of division ; signifying that the former 
f>i the-tvfa ,qa|ntities^^ between which it is placed is to be 
dlvided'by the^ttex . ■«''''•' 

Thus, a-r-6, shows that the quantity represented by a 
is to be divided by that represented by b ; and is read a 
by b, or a divided by h. 

Division is also frequently denoted bj"^ placing one of 
the two quantities over the otlier, in the form of a frac- 
tion. 

h . . 

Thus, 6-4-a and - hoth signify the quotient of h di- 
vided by a ; and -"~- signifies that a - 6 is to be divid- 
a-\-c 

cd by a-f-c. 

= eqxinl to, the sign of equality ; signifying that the 
quantities between which it is placed are equal to each 
ether. 

Thus, x=a -f- h shows that the quantity denoted by x 
is equal to the sum of the quantities a and L ; and is read 
X equal to a plus 5. 

IT identical to, or the sign of equivalence ; signifying 
that the expressions between whieh it is placed are of the 
fame value, for all values of the letters of which they are 
composed. 

* Thus, (.v+a) X. [x — a) znx^-^a-, whatever nunaeral 
values may be given to the quantities represented by x 
and a. 



* Euler calls a; — l=:x — 1 an identical equation; and shows that x is in- 
c^eterminate ; or that any number whatever may be substituted for it:^vhich 
equation is to be expressed according to the present notation ; thus, ar— l_i_a:— 1 . 
In my opinion, Euler's definition of an indentical equation, is preferable to 
Bonnycastle's ; See Euler's Algebra page 289. Vol. 1. 



DEFINITIONS. 3 

> greater than, the sign of majority ? signifying that 
the former of the two quantities between which it is 
placed is greater than the latter. 

Thus, as^h shows that the quantity represented by a 
is greater than that represented by b ; and is read a great- 
er than b. 

< less than, the sign of minority ; signifying that the 
former of the two quantities between which it is placed is 
less than the latter. 

Thus,'az.Z> shov/s that the quantity represented by a 
is less than that represented by b ; and is read a less 
than b. 

: as, or to, and : : so is, the signs of an equality of ra- 
tios ; signifying that the quantities between which they 
are placed are proportional. 

Thus; a : b : : c : d denotes that a has the same ratio 
io b that c has to d, or that a, b, c, d, are proportionals ; 
and is read, as a is to b so is c to d, or a is to 6 as c 
is to d. 

^ the radical sign, signifying that the quantity be- 
fore which it is placed is to have some root of it extract- 
ed. 

Thus, ^a is the square root of a ; ^/a is the cube root 
of a ; and \/a is the fourth root of a ; &.c. 

The roots of quantities are also represented by figures 
placed at the right hand corner of them, in the form of a 
fraction. 

Thus, a2 is the square root of a; a^ is the cube root 
J. 

of a ; and a" is the nth root of a, or a root denoted by 

any number n 

In like manner, a" is the square of a ; a^ is the cube 
of a ; and a'" is the mth power of a, or any power de- 
noted by the number m. 

CO is the sign of infinity, signifying that the quantity 



4 DEFINITIONS. 

standing before it is of an unlimited value, or greater than 
any quantity that can be assigned. 

The coefficient of a quantity is the number or letter 
which is prefixed to it. 

Thus, in the quantities 3b, — |&, 3 and — | are the 
coefficients of b ; and a is the coefficient of x ia the 
quantity ax. 

A quantity without any coefficient prefixed to it is sup- 
posed to have 1 or unity ; and when a quantity has no sign 
before it, + is always understood. 

Thus, a is the same as + a, or -f- la ; and — a is the 
same as — la. 

A term is any part or member of a compound quan- 
tity, which is separated from the rest by the signs + 
or — . 

Thus, a and b are the terms of a + ^ ; and 3a, — 2b, 
and + 5crf, are the terms of 3a — 26 + 5cd. 

In like manner, the terms of a product, fraction, or pro- 
portion, are the several parts or quantities of which they 
are composed. 

Thus, a and b are the terms of ab, or of y; and «, 
i, c, d, are the terms of the proportion a : b : : c : d. 

A factor is one of the terms, or multipliers which form 
the product of two or more quantities. 

Thus, a and b are the factors of ab ; also, 2, a, and 6^, 
are the factors o{2ab^ • and a — x and b — x are the fac- 
tors of the product (a — x) X (6 — x). 

A composite number, or quantity, is that which is pro- 
duced by the multiplication of two or more terms or fac- 
tors. 

Thus, 6 is a composite number, formed of the factors 
2 and 3, or 2X3 ; and 3abc is a composite quantity, the 
factors of which are 3, a, b, c. 

Like quantities, are those which consist of the same 
letters or combinations of letters ; as a and 3a, or dab 
and lab, or 2a^b and 9a''b. 



DEFINITIONS. 6 

Unlike quantities are those which consist of different 
letters, or combinations of letters ; as a and b, or 3a and 
a2, or 5a62 and la'^b. 

Given quantities are such as have known values, and 
are generally represented by some of the first letters of 
the alphabet ; as a, b, c, d, kc. 

Unknown quantities, are such as have no fixed values, 
and are usually represented by some of the final letters of 
the alphabet ; as a;, y, z. 

Simple quantities, are those which consist of one term 
only ; as 3a, bob, — Za-b^ &,c. 

Compound quantities, are those which consist of several 
terms ; as 2a4-6, or 3a--2c, or a+26-3c, &c. 

Positive, or affirmative quantities, are those which are 
to be added ; as a, or -f a, or -f-3a6, k.z. 

Negative quantities, are those which are to be subtract- 
ed ; as —a, or — 3a6, or — lab'^ , &c. 

Like signs, are such as are all positive, or all negative ; 
as + and +> or — and — . 

Unlike signs, are when some are positive and others 
negative ; as + and — ", or — and +. 

A monomial, is a quantity consisting of one term only : 
as a, 26,-3a3^, &c. 

A binomial, is a quantity consisting of two terms ; as 
a-}-b, or a — b ; the latter of which is, also, sometimes 
called a residual quantity. 

A trinomial, is a quantity consisting of three terms, 
as a-\-2b — 3c ; a quadrinomial of four, as « — 26 + 3c — d : 
and a polynomial, or multinomial, is that which has many 

terms. 

The power of a quantity, is its square, cube , biquadrate, 
Sic. ; called also its second, third, fourth power, &c. ; as 
a" , a^, c'*, &c> 

The index, or exponent of a quaLitity, is the number 
which denotes its power or root. 

B 2 



DEFINITIONS. 
Thus, —1 is the index of a- ', 2 is the index of a*, 



I 



and i of a^ or ^a. 

When a quantity appears without any index, or expo- 
nent, it is always understood to have unity, or 1. 

Thus, a is the same as a', and 2x is the same as 2a;"' ; 
the 1, in such cases, being usually omitted. 

A rational quantity, is that v/hich can be expressed in 
finite terms, or without any radical sign, or fi'actional in- 
dex ; as a, or |a, or 6a, Sic. 

* An irrational (Quantity, or Surd, is that of which the 

value cannot be accurately expressed in numbers, as the 

square roots of 2, 3, 5. Surds are commonly expressed 

by means of the radical sign ^ ; as ^^2, ^a, ^/a^, or a 

i — 
fractional index ; as 2^, a^, &c. 

A square or cube number, &c. is that which has an ex- 
act square or cube root, &c. 

Thus, 4 and -^^a"^ are square numbers ; and 64 and 
^\a3 are cube numbers, &c. 

A measure of any quantity, is that by which it can be 
divided without leaving a remainder. 

Thus, 3 is a measure of 6, la is a measure of 35a, and 
^ah of 27 a2&2. 

Commensurable quantities, are such as can be each 
divided by the same quantity, without leaving a remain- 
der. 

Thus, 6 and 8, 2y/2 and 3<^2, ha^ and lah^ , are 
commensurable quantities ; the common divisors being 
2, ^2, and ah. 

Incommensurable quantities, are such as have no com- 
mon measure, or divisor, except unity. 

Thus, 15 and 16, y2 and y3, and a+6 and a^ + 6^, 
are incommensurable quantities. 



* Bonnycastie's definition of a surd was errooeous. Dr. Hutton was led info 
tiie same error, until his mathematics was published in New- York, revised and 
correctf'd by Robert Adrain, LL.D. Professor of Madiematics and JNatural Phi- 
losophy Columbia College. Afterwards, Dr. Hutton published the seventh 
edition of his mathematics, in which he adopted this definition, (which 1 have 
copied from his edition) without mentioning who was the Auiho?: Editor 



DEFINITIONS. 7 

A multiple of any quantity, is that which is some exact 
number of times that quantity. 

Thus, 12 is a multiple of 4, 15a is a multiple of 3a, and 
20a2 6* of 5fl&. 

The reciprocal of any quantity, is that quantity invert- 
ed, or unity divided by it. 

Thus, the reciprocal of a, or -, is „ ; and the recipro- 
cal or v IS -• 

o a 

A function of one or more quantities, is an expression 
into which those quantities enter, in any manner whatever, 
either combined, or not, with knovpn quantities. 

Thus, a-2x,ax-\-3x^, 2x-a {a^ —x^)"^, ax"", a^&c, 

are functions of x ; and axy -f- bx^ , ay -\- x {ax- —hy^)'^, 
&c. are functions of a; and y. 

A vinculum, is a bar , or parenthesis ( ), made 

use of to collect several quantities into one. 

Thus, a -\- b X c, OT {a -{- b) c, denotes that the com- 
pound quantity a + 6 is to be multiplied by the simple 

quantity c ; and ^ab-\-c'^, or (ab-\- 0^)2, is the square 
jroot of the compound quantity afc-j-c^. 

Practical Examples for computing the numeral Valves of va- 
rious Algebraic Expressions, or Combinations of Letters, 

Supposing a=-Q, b-=-b, c=4, c/=l, and c=0. 

Then 

1. a2+2aft — c+c?='36-f 60 — 4-f 1 = 93. 

2. 2a3_3a2fe+c3=432 -540+64 = — 44. 

3. a2 Xa+6 -?a6c=36X 1 1 -240=156. 

1. 2ay/b^ —ac-i-^2ac-\-c2 = 1 2 X 1 +8 = 20. 

5. 3a^2ac+c2, or 3a (2ac-i-c2)^=18^64=144. 

6. ^2a" — ^2ac+c2 =^72- v'64=v'7FI¥=^64 
= 8. 

2a+3c 46c 12 + 12 80 24 80 

7. — ^'._.._j_ — — 1 -» = — _i- — =14, 

6d+4e^^2ac+c2 6+0 ^^48+16 ^^ 8 



S ADDITION. 

Required the numeral values of the following quantities ; 
supposing a, b, c, d, e, to be 6, 5, 4, 1, and 0, respective- 
ly, as above. 

1. 2a2-f36c — 5fZ=127 

2. 5a2& — 10a62-{-2e=--600 

3. 7a2+6-cXd+e=253 

4. 5ya64-62-2afc-e2 = — 7.613871 

o.— Xd-^4-2a2e=A 
c d 



6. 3yc+2av'2a+Z) — d=14 

7. a^a2 +6H-3ic^a2 —62 =245.8589862 

3. 3a26-fyc2+^2acH-c2=:542. 8844991 
g 26+c _ V56+3^c+<^ _, 
*3a-.c 2a+c "* 

ADDITION. 

Addition is the connecting of quantities together by 
means of their proper signs, and incorporating such as are 
like, or that can be united, into one sum ; the rule for 
performing which is commonly divided into the three fol- 
lowing cases* : 

CASE I. 

1 

. When the Quantities are like, and have like Signs. 

RULE. 

/ 

Add all the coefficients of the several quantities together, 
and to their sum annex the letter or letters belonging to 
each term, prefixing, when necessary, the common sign. 



* The term Addition, which is generally used to denote tliis rule, is too 
s-canty to express the nature of ih^: operations that are to be perfonned in it.; 
which are sometimes those of addition, and sometimes subtraction, according 
as the quantities are negative or positive. It should, therefore, be called by 
some name signifying incorporation, or striking a balaace ; iu which case, the 
iticongruitj-, here mentioned, >vould be removed. 





ADDITION. 






EXAMPLES. 




3a 


— 3ax 


26+32/ 


ba 


— 6ax 


^ 66+72/ 


la 


— /ax 


6+22/ 


7a 


— 2ax 


86+t/ 


12a 


— lax 


46+4t/ 


28a 


— I9ax 


206+171/ 


2flj/ 


— ^hy^ 


a — 2x2 


bay 


— Qhy^ 


a— 6as 


Aay 


— fby^ 


4a — x2 


lay 


— Uy^ 


3a— 5a:« 


16ay 


— by^ 
— 18%3 

7a; -4y 


7a — a;2 


24ay 


16(1— 15x» 


Sax'^ 


2a+a:2 


2ax3 


tsc—By ij 


3a+x2 


12aa;2 


3x-y 


a+2x« 


9ax^ 


x-3y 


9a+3cc2 


10ax2 


4x—y 


4ii+x3 


36ax3 


16x-ny . 


19a+8x2 



i) 



CASE II. 
IVhen the Qtiantities are like, but have unlike signs. 

RULE. 

Add all the affirmative coefficients into one sura, and 
those that are negative into another, when there are se- 



10 



ADDITION. 



veral of the same kind ; then subtract the least of these 
sums from the greatest, and to the difference prefix the 
sign of the greater, annexing the common letter or letters 
as before. 



EXAMPLES. 



-3a 
+ 7a 
+ 80 
— a 



2a— 3a:2 
— 7a4-5a;2 
— Za-^-x* 
+a— 3x2 



.3.T-j-2ai/ 
x^Qay 
2x-^ay 



— Ua 


— la * 


6x-{-2ay 


— 2a2 


2ay — 7 


— Sab-^-lx 


Sa"^ 


-ay +8 


+3ab. 


-lOo; 


— 8a2 


+ 2ai/— 9 


-\-3ab- 


-6x 


+ I0a2 


— 3ai/-ll 


— ab- 


-23; 


+ 13o2 


-M2a2/+13 
l-\-12ay-6 


-f2a6- 
H-4a6- 


■|-7x 


+ 10a2 


-4x 


— 2a^a; 


-6a2+26 


Gax" 


4-5:r^ 


+ a^x 


+2a2_36 


— 2ax^ 


— 6x2 


—3a^x 


— 5a2-86 


+3cx2 


-10x2 


+la^x 


+4a2— 26 


-7cx2 


+3x2 


— ia^x 


— 3a2-{-96 


+ 0X3 


+ 11x2 


— a^x 


— 8a2— 26 


4-0X2 


+3x* 



ADDITION. 11 

CASE III. 

When the Qtianiiiies are unlike ; or some like and others un- 
like. 

nuLE. 
Collect all the like quantities together, by taking their 
SUD1S or ditferences, as in the foregoing cases, and set down 
those tiiat are unlike, one after another, with their proper 



signs. 



EXAMPLES. 



bxy 2xy — 2x^ 2aa; — 30 

4ax 3x^+xy 3x^ -2ax 

-xy x'^+xy 5x2—3x2 

-Aax 4x^-3xy 3y.T-flO 



4xy 6x2 -f xy 8^2 - 20 



+ax^ 8a2a;2_3ax 1062-3a2x 

^ax^ la X -bxy -h^-y-^a-x- 

-|-3ax2 9x y — bax 60 -\-Qa-x 



1 



- ax^ 2o2x2-}- xy 0^x2 + 120 



-2ax2 10a2x2+5x2/— ax %~+oa-x'^—a^x-\'\':\ 



+3a2y 2v'x— 18i/ 2a2-.3a^'x 



J. 1 



—2x1/2 3^xy-\-'[Qx x3-2a=.-r'^ 

— 3?/2x ^x-y +25?/ 3a2 — I3x</ 

— ^x~y \^xy.—^xy xy •-r32a2 

+2x1/2 _8i/ +17x'^ 20 —65x2 



oaa?/-3!/2x~8x2i/ |( \2^x + 12x-j/) 37a2 - Sa^^x— ]2x 

y- 

20 



y C \2^x + 12x-j/ ) 37a2 ~ 3a^x— 1 
fi/+10x-i/ ) f,^ 



12 ADDITION. 

EXAMPLES FOR PRACTICE. 

1. Required the sum of i(a-{-i) and ^{a — h). Aus. a. 

2. Add 6x — 3a + 6 4- 7 and — 4a — 3a: + 26 — 9 to- 
gether. Ans. 2x — 7a 4- 36 — 2. 

3. Add 2a + 36 — 4c - 9 and 5a - 36 + 2c - 10 to- 
gether, 'ins. la — 2c - 19. 

4. Add 3a + 26 — 5, a 4- 56 -- c, and Ga — 2c + 3 to- 
gether. Ms. 10a + 76 — 3c — 2. 

5. Add x^ -\- ax^ -j- 6x + 2 and x^ -j- cx'^ + cZa; ~ 1 to- 
gether. Ans.^x^ + {a-\-c) x- + (6+d) x -f 1. 

5. Add &xy — 12x2, — 4x2-f3a:i/, 4a;2 — 2xy, and — 2xy 
+4x2 together. Ans. 4x?/— Sx^-^ 

7. Add 4ax— l30+3xi, 5rr2+3aa;+9x2, 7a;^— 4x2 + 
90, and^x+40 — 6x2 together. Ans. 7ax+8x3 +7xj/. ■ 

8. Add 2a2— 3a6+263— Sa2, 363— 2a2+a3— 5c3, 4c3 
— 2Z>3+5a6+100, and 20a6+l6a2 — 6c— 80 together. 

• Ans. r3a2+22a6+36='+a3— c^+20— 6c. 

oax ^a oa x 

+6(— y^) together. 

. 13a 4c r./bc o6+x^ 

Ans. _^ — 5Vr— 3(-— J-). 

b a T ^ a ' 

10. Add 3a2+46c— e2 + 10, — 5a2+66c+2e2 — 15, and 
— 4a2— 96c— 10e2+21 together. 

Ans. 6c~6a2— 9c2 + 16. 

SUBTRACTION. 

Subtraction is the taking of one quantity from ano- 
ther ; or the method of finding the difference between 
any two quantities of the same kind ; which is performed 
as follows*: 

* This rule being the reverse of addition, the method of operation must be 
-so likewise. It depends upon this principle, that to subtract an affirmative 
quantity from an affirmative, is the same as to add a negative quantity to as 
affirmative. 



SUBTRACTION. IS 

RUtE. 

Change all the signs (-f- and — ) of the lower line, 
or quantities that are to be subtracted, into the contrary 
signs, or rather conceive them to be so changed, and 
then collect the terms together, as in the several cases of 
addition. 

EXAMPLES. 

5a2— 2J a;2-2i/4-3 5x?/+8x— 2 



c> 



Sa^ — n —Sx'^-lly+B 2xy+l6x-\-5 



5xy—\S 8j/2-2^/— 5 10 — Sx-3xy 

-.xy-{-12 — J/2 4-32/-I-2 ^x-{-3-xy 



6xy— 30 9y^^5y—7 7—lx—2xy 



— 5x~y—8a 4^ax~-2x^y i,x^ -{• ■^ x — 4y 

-i-Sx^y — lb 3,yax-5xy^ 6a;2 — Qx- xi 



2 



— .8.^-2 ?/--8a + 76 ^ax — 2x^y-\-5xy^ -x^-^8x+2^x — 4y 



EXAMPLES FOR PRACTICE. 

1. Find the difference of y(«+6) and i(a — J). Ans. b. 

2. From 3a; -2a— 6+7, take 8- 36+a+4a;. 

Ans. 2b'— x- 3a— 1. 

3. From 3a-f64-c-2(Z, take 6-8c+2rf-8. 

Ms. 3a+9c-4d+8. 

4. From 13a;2 -2ax+962, take 5x^—7ax—b^. 

Ms. 8a;2+5aa;+1062. 



That is, — 3« taken from -j-oa, is the same v/ith -f 3a added to +5a=4. 
8a; because, — 3a +3a=0, and + 5o-f-3a=r-f. 8a ; hence, if I take from 
+ 8(1, there remains +8a, but, if the same quantity or equal quantities be 
added to the subtrahend and minuend, their dillerence continues still to be 
thesame as if nothing was added to them : therefore —3o taken tVom -f 5a 
=+8o. And in like manner + 3a taken from — 5a= — 3a — 5a= — 8c. 



14 MULTIPLICATION. 

X 

5. From 20ax-'5^x-\-3a take 4aa;-}-5x2 — a. 

Jlns. 16ax-~l0^x-{-4a. 

6. From5a6+263_c4-6c-.J, take b^ -2ab-\-bc. 

Ans. lab + b^ -c—h. 

7. Fromax^ — &a;2-f-c.x~rf, take bx"-^ex-'8,d. 

Am. ax^~9.bx^-{-{c-e)x-{-d. 

8. From 6a^ 46 ~ 12c + ISx, take 4a; — 9a -f- 46 — 
5c. Ans. 3a + 9.r — 86 — 7c. 

9. From 6x2^ — Sy'^ — ^"J/' *^^^ ^'"^^^ "^ ^(•'^^)2 — 
Aay. Ans. Sa-^?/ — 6^X2/ — Sa?/. 

10. From the sum of 4a.r — 150 4-4x2, 6x-+ 3ax + 
10x2, and 90 — 2ax — 12^x ; take the sum of 2ax •— 
SO + 7x2 , 7xi — 8ax — 70, and 30 — 4^T— 2x2 + 
4o2x2. Ans. llax + 60 — X2 — 4a2x2.. 



MULTIPLICATION. 

Multiplication, or the finding of the product of two 
or more quantities, is performed in the same manner as in 
arithmetic ; except that it is usual, in this case, to hegia 
the operation at the left hand, and to proceed towards the 
right, or contrary to the way of multiplying numbers. 

The rule is commonly divided into three cases ; in each 
of which, it is necessary to observe, that like signs, in 
multiplying, produce +, and unlike signs, — . 

It is likewise to be remarked, that powers, or roots of 
the same quantity, are multiplied together by adding their 
indices : thus, 

1 i 5- 
eXa2, or a'Xa2=a2 ; a-Xa^—a^ ; a'^xa''^=a^ ; and 

The multiplication of compound quantities, is also, 
sometimes, barely denoted by writing them down, with 
their proper signs, under a vinculum, without performing 
the whole operation, as 



MULTIPLICATION. 15 



Sab (a—b), or Sa^as+fta. 
Which method is often preferable to that of executing the 
entire process, particularly when the product of two or 
more factors is to be divided by some other quantity, be- 
cause, in this case, any quantity that is common to both the 
divisor and dividend, may be more readily suppres.-Jed ; as 
will be evident from various instances in the following part 
of the work*. 

CASE I. 

When the factors are both simple quantities. 

RULE. 

Multiply the coefficients of the two terms together, 
and to the product annex all the letters, or their powers, 
belonging to each, after the manner of a word ; and the 
result, with the proper sign prefixed, will be the product 
required t- 



* The abore rule for the signs may be proved thus : If" b, b, be any two 
quantities, of which b is the greater, and b — 6 is to be multiplied by a, it is 
plain that the product, in this case, must be less than ob, because b — bis less 
than B ; and, consequently, when each of the terms of the former are multi- 
plied by a, as above, the result will be 

(b — 6)Xa=aB — ab. 

For if it were as + ab, the product would be greater than as, which is ab- 
surd. 

Also, if B be greater than b, and a greater thana, and it is required to mul- 
tiply B — 6 by A — a, the result will be 

(b — 6)X (a — a)=AB — as — 6a 4. 06 

For the product of b — b by a is a (b — b), or ab — a6, and that of b — 6 
by — a, which is to be taken from the former, is — a(B — b), as has been al- 
ready shown ; whence b — b being less than b, it is e% ident that tJie part 
which is to be taken away must be less than as ; and consequently since the 
first part of this product is — as, the second part must be 4-o6; for ifit were 
— ab, a greater part than ob would be to be taken from a(b ^^ 6), which is ab- 
surd. 

f WTien any number of quanthies are to be multiplied together, it is the 
same thing iii whatever order they are placed : thus, if 06 is to be multiplied 
by c, the product is either abc, acb, or bca, &c. ; though it is usual, in this case, 
as well as in addition and subtraction, to put them according to their rank in 
the alphabet. It may here also bj observed, in conformity to the rule given 
above for the signs, that ( + «) v ( + 6), or ( — a) X ( — b)=+ab ; and ( -f-e) 
X (.-'"), or (-o)X( + 6)= -a6. 



IS 



12a 
36 


1 
MULTIP 

EXAN 

-2a 
+46 


LIGATION. 

!PLE9. 

-{-5a 

-6.r 

— 30ax 


— 9x3 

— bhx 


36ab 


-8c& 


+456^3 


7a6 

— 6ac 


-.6a2je 
+5a; 


-2a:^2 
—xy 


— laxy 
+6ay 


35a3fcc 


-30a3x2 


+2x2?/ 3 


— 42a'' xy 


3aib 
26a9 


]2as.r 

— 2x 1/ 


— 6rc^z 


— a^xy 
+2x1/* 


6a b 


-24a2a:3y 


— Ga.rj/^r^ 


— 2a2x3j/5 



CASE 11. 
W%e» one of the factors is a compound quantity . 



RULE. 



Multiply every term of the compound factor, consider- 
ed as a multiplicand, separately, by the multiplier, as in 
the former case ; then these product.*, placed one alter 
another, with their proper signs, will be the whole pro- 
duct required. 

EXAMPLES. ♦ 

3o— 26 6xy— 8 a2-.2x + l 

4a 3x 4x 



12aa— 8a6 nx^y^24x " 4a^x-8x3+4.c 



MULTIPmCATION. 17 

6a ^ -2x 3^2/ 



60aa:-5a3 6 —70a: +14ax Sx^^+ary^ -2xy 



13x2- a26 SSxiz+Sa" 3x2_xy— S^/S 

—2a 13.T2 5x2 , 



— 26ax2+2a36 325x3i/+3ea2x2 I5x*—5x^y - I0x^y2 

CASE III. 
FF/te» both the factors are compound quantities. 

RULE. 

Multiply every term of the multiplicand separately, by 
each term of the multipUer. setting down the products 
one after another, with their proper signs ; then add the 
several lines of products together, and their su» will be 
the whole product required. 

EXAMPLES. 

x+y ox+4y x^+xi/— y 

x+y 3x— 2y X — y 



x^-}-xy lox^-\-\2xy x^+x^^ — xy^ 

+xy+y^ — lOry Sy^ —x^y-xy^-^-y^ 

x^-\-2xy+y' 15x2+ 2xy-Sy^ ' x^ * —2xy^-\-y^ 

x-\-y x^-\-y x2-|-xy+3/2 

x—y x^-^y X —y 



x^+xy x^+x^jf x^4-x2y-f-xi/2 

--Xy—y'' +x2y + J/2 ~x2^/__Xi/2_2/3 



t2 * 



- 2/2 a;4^2x22/+y2 -VpS ♦ * „^3 



c 2 



38 MULTIPLICATION. 



EXAMPLES FOR PRACTICE. 

1. Required the product of a;2 —x^-|-3/2 andx-\-y. 

Ans. x^-\-y^. 

2. Required the product of x^-\-x^y-\-xy^-^y^ and 
re— 2/' ^^ns. x*—y*. 

3. Required the product of x^-\-xy-{-y^ and x^ — xy-\- 
y^' An$. x^+x^y^-^y*. 

4. Required the product of 3a;2__2xj/+5, and x^ +2a^/ 
--3. Ans. 3x«4-4x3j/-4a-22/2— 4a;2 + 16a2/- 15. 

6. Required the product of 2a^ — 3aa;+4a-2 and ba^ — 
6ai-2x2, Ans. IOC --27a3a:+34a2a;2 — IBax^ - S.-c". 

€. Required the product of 6x3-f-4aar2 4-3a2a;4-a3, and 
2x2 -3aa;-f-a2. j9mj. lOx* — Vax^ — a^x^ -Sa^ais-f «* 

7. Required the product of 3x^-\-2x^y2-\-3y^ and 2x3 
»-3x» 1/3 +52/3. 

j3ns. 6x6 — 6x*i/' —Qx* y'^ ^'ilx^y^ +x^y^ -{-Ibx* . 

8. Required the product of x^ — ax~^bx—c and x^ — 

t?x4-e. ^ns. x^— ax^ — cix^ + (64-ad+e)x3 — (c+^ci+^O 
x^ + (cd-\-eh)x—ce. 

9. t Required the product of the four following fact- 
lors, viz. 

L II. III. IV. 

{a + b) (a8 +a6+.62) {a — h) and (a^ — a6 + i^), 

Ans. a6 — fcs^ 

10. Required the product of a^ 4" ^a^oc + 3ax2 + x^ 
and a3 — 3a2x + Sax^ — x^. 

Ans. a^ — .3a4x2 + SflSa* — x«. 

11. Required the product oi a'^-\-a'^c^-{-c'^ anda^ — c^. 

Ans. a* — c". 

12. Required the product of a^ •\- h^ -\- c^ — ah — ac 
— he and a -\- b -\- c. Ans. a^ — Sate -\- h'^ -{- c^. 



f I would advise the learner to perform the calculation of this example 
several ways, viz. First, by multiplying the product of the factors I. and 
II. by the product of '.he factors IIL and IV. Secondly, by multiplying the 
product of the factors I. and III. bj' the product of the factors II. and IV. 
Thii-dly, by rrubiplying the product of the factors I. and IV. by the product 
of the factors 11. anS III. The last method is the most concise ; See Euler'b 
Algebra page 119. Vol. I. 



DIVISION. 19 

DIVISION. 

Division is the converse of multiplication, and is per- 
forrneJ like that of numbers ; the rule being usually 
Jitided into three cases ; in each of which like signs give 
+ m the quotient, and unlike signs — , as in finding their 
products*. 

It is here also to be ohserved, that powers and roots of 
the same quantity, are divided by subtracting the index of 
the divisor from that of the dividend. 

Thus, a3-j-a2, or-t=a ; 0=^-7-0% or,-S=a^ ; 

J ' 4' or— =a>2 ; and a"'-^a". or — —a'^-^ 

CASE I. 

When the divisor and dividend are both simple quantities, 

RULE. 

Set the dividend over the divisor, in the manner of a 
Taction, and reduce it to it simplest form, by cancelling 
he letters and figures that are common to each term. 

EXAMPLES. 

Gab 12ax^ 

6a6-r-2a, or-^— =36 ; and ISax^-j. 2x, or— „ — =4ax ; 

a a 
i-i-a, or -= 1 ; and a -f- — a, or = — 1. 



* According to the rule here given for the signs, 1t follows that 
-f-a6 — ah — ah_ •\-ah 

Is'will readily appear by multiplying the quotient by the divisor j the signs of 
le products being then the same as would take place in the former rule. 



'20 DIVISION. 

— 2a ' 111 

Also - 2a -r 3a, or -^ — — | ; and 9x^ -^3x* =3x* , 

1. Divide IGx^ by 8.x, and 12a'' x'^ by -Qa^x. 

3x 
Ans. 2x, and——* 

2. Divide — 15ay^ hy Say and — IQax^y by — 8ax. 

Ans. — oy, and i 

SrDivide— -a *, by Ta2,andax3 by — -a^x*. 

^/is. _31, and -^^a^xT^- 

2 X 

4. Divide na-b'^ by — 30=6, and — Ibay^ by — 3ay^. 

Ans. — 46, and dyh 

5. Divide — Iba'^x- by 5ax^ , and 21 o^cs-j-i by — 7a 

c^xT. Ans. — 3a, and — 3ax4. 

6. Divide — llx^a^c by — 5x^a~c2^ and 24 ^xy by 

— Ana Hxiaei 

»yx?/. ^"^- ^~»and3yxy. 



CASE II. 

When the divisor is a simple quantity, and the dividend a : 
compoimd one. 

RUiE. 

Divide each terra of the dividend by the divisor, as in 
the former case ; setting down such as will not divide in 
the simplest form they will admit of. 

EXAMPLES 

a6+62 a+b 

{ab+b^) -f. 26, or-^ = ^a+^b =-^ 

10a6— 15ax 
(10a6— 15ax) — 5a, or =26 — 3x. 



DIVISION. 21 

30ax-4Bx- 
(30ax-48x9) ^ Qx, or -^- = ba *~ Bx 

1. Let 3x3+6x2 4-3rtar—15.T be divided by 3x. 

Ans. a;2+2x+a--6. 
s2. let 3abc-\-12ahx — 9a^b be divided by 3ab. 

Ans. c-\-4x—3a. 

3. Let40a363-{-60a2fe2_i7aJ be divided by -ab. 

Ans. -.40a^b2—60ab+n. 

4. Let ISa'bc —12acx'^-\-dad^he divided by -buc. 

12x2 d2 

Ans. — 3ab-] — . 

5 c 

5. Let 20ax3-f 15a.x' + 10ax+6a be divided by 5a. 

Ans. 4x3 4-3.x2-{-2x+l. 
C. Let 6bcdz -4- 4bzd^ — 262^^2 be divided by 2bz. 

Ans. 3cd + 2(/2 — bz. 

7. Let 14a2 — lab + 21ax — 28o be divided by 7a. 

Ans. 2a — b -\- 3x — 4. 

8. Let — 20ab -f 60ah^ — 12a262 be divided by — 4ab. 

Ans. 5— 1562 _|. 3ab. 

9. Let Iba^bc — 12acx2 + bad- be divided by — 5ac. 

, . 12x2 ^2 
Ans. — Sab H • 



CASE III. 

When the divisor and dividend are both compound 
quantities. 

RULE. 

Set them down in the same manner as in division of 
numbers, ranging the terms of each of them so, that the 
higher powers of one of the letters may stand before the 
lower. 

Then divide the first term of the dividend by the first 
term of the divisor, and set the result in the quotient, 
ivith its proper sign, or simply by itself, if it be affirma- 
i ive. 



22 DIVISION. 

This being done, multiply the whole divisor by the 
term thus found ; and, having subtracted the result from 
the dividend, bring down as many terms to the remainder 
as are reqnisite^for the next operation, which perform as 
before ; and so on, till the work is finished, as in cgmmon 
arithmetic. 



EXAMPLES. 



x'^-\-xy 

xy+y^ 
xy-\-y- 






4a''x-\-iax^ 



ox*+x3 



a;-3)x3-9x2+27x-27(x2-6a;+9 
x3 — 3-x2 



— 6x2 4-:^7x 
— 6x2 + 1 8x 



9x-27 
9x—27 



DIVISION. 23 

1, 

2.x»— 3ax-}-a2)4x*— 9a=cc2_{_6a3a;-a^(2a;3+3aa;-a= 
4a;4— 6aa;3+2a3i2 



6ax^ - 9a2a;2-f3a3a; 

-2a2xa+3a'a— a* 
_2a2x2-j-3a3x~a* 



Note 1. If the divisor be not exactly contained in the 
dividend, the quantity that remains after the division is 
finished, must be placed over the divisor, at the end of 
:he quotient, in the form of a fraction : thus*. 



0/V.3 

a+x)a3-a;3(a2— ax + i" 



a-\-x 



— a~x — x^ 

— a^x— aa;2 

ax^-\-x^ 



* In the case here given, the operation of division may be considered as 
rminated, when the highest power of the letter, in the lirst or leading ternti 
the remainder, by which the process is regulated, is less than the power 
the first term of the divisor; or when the first term of the divisor is not 
imtained in the first term of the remainder ; as the succeeding part of the 
lotient, after this, instead of being integral, as it ought to be, v/ould neces- 
rily become fractional 



24 DIVISION'. 

—x^y+y'^ 



x^y^-\-xy^ 

-X2/3 4-1/4 

—xy^—y* 

2y* 
2. The division of quantities may also be sometimes 
carried on, ad infinitum, like a decimal fraction ; in which 
case, a few of the leading terms of the quotient will gene- 
rally be sufficient to indicate the rest, without its being 
necessary to continue the operation ; thus, 

/ ^ a a^ a^ a* 

a'\-x 



— X 

— X 



a 



X3 g 



* Now, it is easy to perceive tliat the next or 6th term of the quotient will 

fee — — , and the seventh term , and so on, alternately /)i«s and minus ; 

as a6 



DIVISION. 2(5 

And by a process similar to the above, it may he shown 

that 

a X a-2 x^ X* x' 

Where the law, by which either of these series may be 
continued at pleasure, is obvious*. 



this is called the law of continnaiion of the series. And the sum of all the 

terms when infinitely continued is said to be equal to the fraction — _ . 

a+a? 

Thus we say the vulgar fraction _ when reduced to a decimal is =22222, &c. 

infinitely continued. The terms in the quotient are foimd by divrding the re- 
mainders by o, the first term of the divisor ; thus, the first remainder — x di- 
vided by o, gives — ^ the second term in the quotient; and the second re- 
ft 

niainder + — divided by a gives + — the third term, &c. 
a a2 

* In this example, if a; be less than a, the series Is convergent, or the value 
of the terms continually diminish ; but when a; is greater than a, it is said to 
diverge. 

To explain this by numbers : suppose a = 3, and x = 2. 

Then, H-i . ^-i-r^, &c. 
a ' as a2 

The corresponding values are, 

^ ^3~9~27 
■R'herc the fractions or terms of the series grow less and less, and the farther 
tliey are extended, the more they converge or approximate to 0, which is 
supposed to be the last term or limit. 
But if a=2, and x=3. 

X x2 xs . 

Then, 1 +-+-:r+-T' ^^' 

' ' a ' a2 ' a3 

The corresponding values are, 

3 ,9 .27 .^ 
•1 I — 1_— J — , &c. 

^2 M~8 

In which the terms become larger and larger. This is called a diverging 

series. 

If a;=l, and a=l in the preceding example : 

m a _ X , xn x3 ^ .,, , 1 

Then, = i \. ^ &c. will be— — -=l_l J.l_l, &c. 

Now, because =2", it has been said that 1 — 1~}"1 — 1, &c. infinitely con- 

14-1 

tinued, is =■!■ : a singular conclusion, when it is perceived from the terms 

themsehes, that their sum must necessarily be either or -^-l, to whateret 

D ' 



26 DIVISION. 



EXAMFLES FOR PRACTICE. 

1. Let g2 — 2oa:+a;2 be divided by a—x. Ms. a— cc, 

2. Let x^—3ax2-\-3a~x—a^ be divided by a; -a. 

Jlns, x2— 2aa;+a2. 

3. Let a3+3a2a;+5aa;2+x3 be divided by a+x. 

Ans. a2+4ax+x2. 

4. Let 2i/3 -192/3-f 26j/— 16 be divided by y— 8, 

Ans. 2if—3y-\-2. 

5. Let xs 4-1 be divided by cc+l, and x6 — l by x-1. 
.^ns. x^ — x3 +x2 - x+ 1 , and x* +a« +x3 -rx^ +x+ 1. 

6. Let 48x3 — 76ax2 — 64a3x4-105a3 be divided by 2x 
— 3a. .4n«. 24x=-2tJX— 35a2. 

7. Let 4x4 — 9x3-f-^x-l be divided by 2x2+3x—l. 

.3ns. 2x2— 3x4-1 • 

8. Let X*— a'x2-|-2a3x — a^ be divided by x2 -ax4-a^. 

Ans. x^-\-ax — a^. 

9. Let 6x* — 96 be divided by 3x— 6, and a^+x* by a 

..^ws. 2x3+4x2 + 8x4-16, and a"— a3x+o2x2-ax3+x''. 

10. Let 32x5 4-243 be divided by 2x+3, and x^ — a« by 

Ans. 16x*— 24x3+36x2— 54X+81, andx5+a:*;* + 
a2x3+a3x2+a*x+a5. 

11. Letfc«--3?/4 be divided hy b—y, anda*+4a2&+ 
Si by a + 26. 

Jns. 63+52y+jy2 4-y3__2_, anda3-2a26+4a6 

o — 2/ 

1663+246* 
+4a62_862 _863+__^-^-. 



-xtent the division is supposed to be continued. The real (juestion, however, 
results from tlie fractional parts, which (by thedivison) is always + ^ when 
the sum of the terms is 0, and — i when the sum is + 1 : consequently i 
IS the true quotient in the former case, and 1— i in the other. Editor. 



ALGEBRAIC FRACTIONS. 27 

12. Let x3-f-p.r;+9 be divided by x+Oj and x^ — px^-^ 

■]X-'—r by x—u. 

Alls. x-\-p^a-\-- — —. , and x"-^ (a'-p) x-ap 

* x-f-a 

x—a 

13. Let 1— 5x+10x3 — 10x3+5x* — x^ be divided by 
l-2x+x3. Ms. 1— 3x+3x2— x'. 

14. Let a^ 4-46* be divided by a2_2a6-|-263. 

15. a^ — Sa^x+lOa^x^ — lOa^x^-f-^ax*— xs be divided 
bya2-2ax+x2. Ans. o^ — Sa^x-f-SaxS — x^. 

16. Let a^-ffc" be divided by a''+ab^2+b^. 

A71S. a^ -dh^2-\-i^- 



OF ALGEBRAIC FRACTIONS. 



Algebraic fractions have the same names and rules of 
operation as numeral fractions in common arithmetic ; and 
the methods of reducing them, in either of these branches, 
to their most convenient forms, are as follows : 

^ CASE I. 

Tojlnd the greatest common measure of the terms of a frac- 
tion. 

RULE. 

1. Arrange the two quantities according to the order of 
their powers, and divide that which is of the highest di- 
mensions by the other, having first expunged any factor, 
that may be contained in all the terms of the divisor, with- 
out being common to those of the dividend. 

2. Divide this divisor by the remainder, simplified, if 
necessary, as before ; and so on, for each successive re- 
mainder and its preceding divisor, till nothing remains, 



28 



ALGEBRAIC FRACTIONS. 



when the divisor last used will be the greatest common 
measure required ; and if such a divisor cannot be found, 
the terms of the fraction have no common measure,* - 

Note. If any of the divisors, in the coarse of the ope- 
ration, become negative, they may have their signs chang- 
ed, or be taken affirmatively, without altering the truth of 
the result ; and if the first term of a divisor should not be 
exactly contained in the first term of the dividend, the se- 
veral terms of the latter may be multiplied by any number, 
or quantity, that will render the division complete!. 

EXAMPLES. 

1. Required the greatest common measure of the frac- 



tlOD 



'-{-a:' 



X^ — X 



X 



■■+x 
or a;2-f 1 



z*-{-x^ 



- .C2 — I 

—x^ — 1 



* If, by proceeding in this manner, no compound divisor can be found, that 
is, if tlie last remainder be only a simple quantity, we may conclude the case 
proposed does not admit oi" any, but is already in its lowest terms. Thus, for 



instance, if the fraction proposed were to be ~ ■ — '- ; it is 

plain by inspection, that it is not reducible by any simple divisor", but to 
know whether it may not, by a compound one, I proceed as above, and find 
the last remainder to be the simple (juantity 7.i2 : whence I conclude that 
the f)-action is already in its lowest terms. 

t In findin,'j the greatest common measure of two quantities, either of them 
may be multiplied, or divided, by any quantity, which is not a divisor of the 
other, or that contains no factor which is common to them both, \» itiiout in any 
respect changing the re-sult. 

It may here, also, be farther added, that the common measure, or divisor, 
of anj number of quantities, may be determined in a similar manner to that 
given above, by first finding the common measure of two of th'im, and then 
of that common measure and a third ; and so on to the last. 



ALGEBRAIC FRACTIONS. 2^ 

Whence x^ + l is the greatest common measure required. 
2 I'equired the greatest common measure of the frac- 

tion — ili 

x3+26j:2-f.62a; 

*_ 26x2 _ 262x1 

orx-i-6 I a;2+26x-f-62(.T+6 
a;2 -{-6a; 



Z>a;4-62 
6a; 4-63 



Where x-f-6 is the greatest common measure required. 

3. Required the greatest common measure of the frac- 

.- 3a2-2a— 1 

tion • 

4rt3 _2a2— 3a+l 

3^2— 2a- l)4a3-2o2~3a4-l 
3 



12a3_6a2-9a+3(4a 
12a3 — 8a2 -4o 



2a2 - 5a4-3)3a2 -{-2a- 1 

2 



6a2_4a-2(3 
6a2_15a-}-9 



11a — 11 ora— 1 



Where, since a— l)2a2- oa-{-3(2a— 3, it follows that the 
last divisor a— 1 is the common measure required. 



» Here, I divide the remainder— 26a:2 —262 x by— 2xb, (its greatest sim- 
, pie divisor) and the quotient is x-f-b ; and then 1 divide" the last divisor by 
' x+b, Sic. Editor. 



30 ALGEBRAIC FRACTIONS. 

In which case the common process has been interrupted 
in the last step, merely to preventthe work overrunning 
the page. 

4. It is required to find the greatest common measure 

offlmf-.. Ans.x — a. 

a;4 o* 

5. Required the greatest common measure of the fracr 

tion ^':=f ' '^«^- «^ -^' • 

a^.~.a^x — ax2-{- x^ 

6. Required the greatest common measure of the frac- 
tion x^+a'x^+a' Ms. x^+ax+a^ . 

x*-{-ox^ -a3x— a* 

7. Required the greatest common measure of the frac- 

7a2— 23a6 + 663 . or 

tion . ■- Ji/is. a — oo. 

8. ^Required the greatest common measure of the frac- 
tion ^J±^±^^jzl^l5±^''^. Ms. x-f 2a. 

a-2 —bx-j-2ax—2ab 



* This fraction can be reduced bv Simpson's ru'e (page 50) thus : 
Fractions that have in them more than two diil'erent letters, and one of the 
letters rises only to a single dimension, either in the nujnerator or denomma- 
lor, it will be best to divide the said numerator or deiiomniator (whichever 
it is) into two parts so that the said letter may be found m every term o4 the 
one part, and be totally excluded out of the other : this bemg done, let the 
"■reatest common divisor of these two parts be found, which will evidently be 
Z divisor to the whole, and by which the division of the quantitj- is to be tried ; 
as in the following example," where the fraction given is 
a3-^ax 2 ^h:'2—2a2 x-\-bax-2ba2 

x2 — bx-^2ax — 2ab 
Here the denominator being the least compounded, and 6 rising therein to a 
siingle dimension onlv, I divide the same into the parts x2 -}-2a,r, and — t-^— 
2at ; which, by inspection, appear to be equal to (a:-t-2a) X a:, and {x-i-ja) 
X— t Therefore .T-i-2a is a divisor to both the parts, and likewise to the 
whole, expressed by (x-f-2«) X (cc~b) ; so that one of these two factors, it 
the fraciion given can be reduced to lower terms, must also measure the nu- 
raertuor: but the former will be found to succeed, the quotient coming out 
X2 —ax -^ bx — ab, exactly : whence the fraction itself is reduced to 
x2—a x-^bx—ab^ which is not reducible fartlier by x— 6, since the divi- 
sion does not terminate without a remainder, as upon trial will be found. 

This rule is sometimes of great utility, because it spares ^eat labour, and 
is very exffeditious in reducins several fractions. ' £.aito) ■ 



ALGEBRAIC FRACTIONS. 31 

9. Required the greatest common measure of the frac= 

a;*— 3a.T3— 8g2 a;2 + 5 8a3.r— 8<J-* . 

don — r— — r— — . Ans.a;2+2ax— 2a2. 

x^ — ax~ —Ua'x-\-ba^ 

10. Required the greatest common measure of the frac- 

5Q« + 10a464-5(/3i,2 

lion — ;— ; — ■ -_——_, Ans. a-\-b. 

a36+i;a262+2a63+Z** 

11. Required the greatest common measure of the frac- 

tion ; ;r-n TT-. i -. — • Ans. Sa^ — 2c3, 

" 9a36— ^7a2 6c— 6a6c2-|-l86c3 

CASE II. 

To reduce fractions to their lovcest or most simple terms. 

RULE. 

Divide the terms of the fraction by any number, or 
quantity, that will divide each of them without leaving a 
remainder ; or find their greatest common measure, as in 
the last rule, by which divide both the numerator and de- 
nominator, and it will give the fraction required. 

EXAMPLES. 1 

1. Reduce and — — — to their lowest terms. 

Here ^h==£^ Ans. And -^—^-^ Ans. 
ba^b^ bb ax+x^ a-\-x 

2. It is required to reduce-ffltf to its lowest terms. 

a^c-\-a-x 



Here cx-^x'- 
or c-\-x 



a^c^a-x 

a^c+a^x{a^ 

a^c+a^x 



Whence c+r is the greatest common measure ; 
and c4-x) '. — — — the fraction required. 



32 ALGEBRAIC FRACTIONS. 

3. It is required to reduce — 1-7- . to its lowest 
terms. 



— 26a;2-262a; 
or x-^b 



x2+2bx+b^(x+b 
a-2 +bx 



bx+b^ 
6x+fe2 



Whence x+b is the greatest common'measure ; and x-f 6) 

. ~ ;= — Z! — the fraction required. 

a;a+2/JxU2 x+b 

And the same answer would have been found,if tc^— i^^; 
had been made the divisor instead of x^-\-2bx+b^. 

x^ - a^ 
4. It is required to reduce to its lowest terms. 

x^ — a^x^ 

Ans. '. J" — . 

6. It is required to reduce ^ -; to its lowest 

^ , 6a2-|-llax+3x2 

terms. ^Sa — x 

3a-\-x 
2-j;3 i6x— 6 

6. It is required to reduce — - to its low- 

est terras. .^«s. f. 

„ ... ... , 9;c5_[.2x3 + 4x2— r + 1 

7. It IS required to reduce ; - 

16x* — 2x3-fl0x2 — x+2 

to its lowest terms. a Sx^-fx-' + l 

6x2+x+2* 



ALGEBRAIC FRACTIONS. 33 

a2fj2_c3(j3_ a2c2 4.c4 

8. It is required to reduce ^^^ d—iacd—2ac-- -]-2c'^ ^^ 
its lowest terms. Ans. iorf^H^c^ ' 

■! CASE III. 

'J'o reduce a mixed quantity to an improper fraction. 

RULE. 

Multiply the integral part by the denominator of the 
fraction, and to the product add the numerator, when it is 
affirmative, or subtract it when negative ; then the result, 
placed over the denominator, will give the improper frac- 
tionrequired. 

EXAMPI<ES. 

1. Reduce 3| and a to improper fractions. 

^ ^^ 3X5+2 15+2 17 

Here3|= — =—-—=— Ans. 

* 5 5 5 

^ , h aXc — b ac — b a 

And a = = •^»*- 

c c c 

2. Reduce x-\-~ and x to improper frac- 

X X 

tions. 

, a xYsX-^-a x^-\-a . 
Here x + -= = Ans. 

XX X 

, , ^ a^—x- a:2— fl2+x2 2x2— a^ 
And a* = = Ans. 

XXX 



* xy,x=x2. In adding the numerator a 2 — x2 , the sign — affixed fu 
the frnrtini /'- ^" , denotes that the whole of that fraction is to be subfract- 

X V 

ed, and consequently that the signs of each term of the numerator must be 



34 ALGEBRAIC FRACTIONS. 



2a; 

3. Let 1 be reduced to an improper fraction. 

Ans. 

a 
3x — b 

4. Let 6a — be reduced to an improper fraction. 

- Ans. . 

a 

o. Let a: — be reduced to an improper fraction. 

^" „ 2ax—a—x^ 

Ans. , 

2a 

2a; — 7 

6. Let 5+ — - — be reduced to an improper fraction. 

^^ ^ 17i-7 

Jins. 

3a: 

7. Let 1— be reduced to an improper fraction. 

a 

x—3 

8. Let I+2j: be reduced to an improper frac- 

ox 

tion. ^ 10x2 4-4x-|-3 

Ans. . 

CASE IV. 

To reduce an improper fraction to a whole or mixed 

quantity. 

RULE. j 

Divide the numerator by the denominator, for the inte-! 
gral part, and place the remainder, if any, over the deno-, 
minator, for the fractional part ; then the two, joined to- 
gether, with the proper sign between them, will give the 
mixed quantity required. 



« 



changed wl»en it • is combined with X2 , hence the improper fraction is 

X 



xa — as -J-Xg -2x2 — o3 
' — ^ or — 



ALGEBRAIC FRACTIONS. 35 



.1 



01: 



EXAMPLES. 

1. Reduce — and °^"^"- to mixed quantities, 

27 
Here _=27^5=5| Ans. 
5 ^ 

And =(ax-j-a2)^a:=a+ — Ans. 

X ^ ^ X 

2. It is required to reduce the fraction to a 

X 

whole quantity. Ans. a— a:^, 

3. It is required to reduce the fraction — to a 

ab 

mixed quantity. ^^^^ j 2a 

4. It is required to reduce the fraction — to a 

U'-X 

(nixed quantity. Ans.a+x+^, 

a — x 

fv* 3 ,^ qj 3 

5. It is required to reduce the fraction — to a 

vhole quantity. Ans. x'^+xy-^-y'^. 

]0x^ - 5.t+3 

6. It is required to reduce the fraction 

ox 

3 
a mixed quantity. Ans. 2x — H . 

OX 

CASE V. 

fo reduce fractions to other equivalent ones, that shall have 
a common denominator. 

RULE. 

Multiply each of the numerators, separately, into all the 
ienominator?, except its own, for the new numerators, and 



36 ALGEBRAIC FRACTIONS. 

all the denominators together for a common denominu 
lor*. 

EXAMPLES. 

1. Reduce - and - to fractions that shall have a common 
c 

denominator. 

^^"^^^^^^^o ( the new numerators. 



bxc=bc the common denominator. 

Whence, t and -=7- and 7- > the fractions required. 
' o c be be 

2x b . 

2. Reduce — and -to equivalent fractions having a com- 

a c a u 

"ex 0.0 
mon denominator. ji^g — and — • 

ae ac 

3. Reduce - and — ^ to equivalent fractions having a 

_, ^ . , " , ac ab+b"- 

common denominator. Jins. — and — ; — 

be be 

3r 26 

4. Reduce—, ^:^, and d, to equivalent fractions having 

2a oc Q^^ ^^^j Qacd 

a common denominator. ^^s. ,5 — a — and 



Qac'' 6ac 6ac 

3 2.r 4x 

5. Reduce -, — and a -j — , to fractions having a com- 
4' 3 ^5 

mon denominator. 45 AQx 60a-f 48x- 



Ans. rr: , ^TT- and 



60 ' 60 60 



* It may here be remarked, that if the numerator and denominator 
a fraction be either both multiplied, or both divided, by the same numbe: 
or quantitj', its value will not i3e altered : thus 

2 2X3 6 3 3-r3 1 a ac ab a 

3 3X3 9' 12 12-7-3 4' 6 6c be c 
which method is often of great use in reducing fractions more readily t 
a common denominator. 



ALGEBRAIC FRACTIONS. 37 

b. Reduce -, — , and > to iractions having a cona- 

.27 a—x 

mon denominator. la^ —lax 6ax—6x'' I4a-f.l4.r^ 

"*■ 14a- Hrc' 14a- 14a;^ Ha-Ti^' 

CASE VI. 

7% add fractional quantities together. 

RULE. 

Reduce the fractions, if necessary, to a conimon deno- 
minator ; then add all the numerators together, and under 
their sum put the common denominator, and it \yill give 
the sum of the fractions required*. 

EXAMPLES. 

X • % 

1. It is required to find the sum of— and -. 



i=3x I 
l=z2x I 



Here "^v^o. o'*^ ^ the numerators. 



And 2x3=6 the common denominator. 

3x 2x 5x 
Whence if ■T"'7r=-7-, the sum required. 



2. It is required to find the sum of-, -, and ? 

b d f 

Here aXdXf=adfi 

cX b'><f=cbf \ the numerators. 

eXbXd=ebdS 



• In the adding or subtracting of mixed quantities, it is best to bring 
the fractional parts only to a coramon denominator, and then to affix 
their sum or difference to the sum or difference of the integral parts, in- 
terposing the proper sign. 

E . 



38 ALGEBRAIC FRACTIONS. 

And b X. d xf=bdf the common denominator. 
adf cbf ebd adf-\-cbf-j-ebd 
Whence ^^+^-^^+^^= bdf ^^^ '""^^ 

3 X' 'Zax 

3. It is required to find the sum of a — and 6+ — 1^. 

^ be 

Here, taking only the fractional parts, 
We shall have J 2axXb=2abx \ ^^e numerators. . 

And bXc=hc the common denominator. 

3cx2 2abx 2abx - Zcx^ 
Whence a — —4.6^-7—= a-\-h-\- r the sum. 

4. It is required to find the sum of -^ and -r- 



Ans. — . 
35 



5. It is required to find the sum of — and -. 



. 15a;+2a.T 
Jins. 



10a 



6. It4s required to find the sum of -, ~, and - 



Ans. — . 
12 



4a; a;— 2 

7. It is required to find the sum of — and 



7 5 

27.T-14 



Ans. 



35 

2x 8a; 

8. Required the sum of 2a, 3a -| and a — — • 

5 9 

22x 

Ans. 60 . 

45 

. ' 3x a a—x 

9. Required the sum of 2a-f— , , and • 

Sa^'x — Sax^+bx^ 



Ans. 2a+2+ 



5a* — 5ax 



ALGEBRAIC FRACTIONS. 33 

10. Required the sum of 6a;H — — and 4a; — —r—-. 

5x2 — 16.r+9 



Ans. 9x- 



Ux 



2a ,a-f-2a; 
11. It is required to find the sum of 6a;, --^» and— — 

.ins.5x-^ _- -. 

CASE VII. 

To subtract one fractional quantity from another. 

RULE. 

Reduce the fractions to a common denominator, if ne- 
cessary, as in addition ; then subtract the less numerator 
from the greater, and under the difference write the com- 
mon denominator, and it will give the difference of the 
fractions required. 

EXAMPLES. 

2x Sx 

1. It is required to find the difference of -^ and — • 

Here |^>^|ZJ^^ I the numerators. 

And 3X6=15 the common denominator. 

1 Ox 9 z" X 
Whence f^— 77=77, the difference required. 

x—a 

2. It is required to find the difference of —7- and 

2a— 4a; '^^ 



3c 

TT (x — a") X3c=3c.T— 3ac ) ., . 

"^42a~.4r)X26=:4a6-86x ] '^' ""^^rator. 



40 ALGEBRAIC FRACTIONS. 

And 26x3c=6&c the common denominator. 
2,cx — oac Aab — Ux ^cx — 3ac — 4a6 + ^^^ 
Whence —^——~Gb^^ 667 

the difference required. 

3. Required the difference of — — and — • Ans. •''c+ol' 

l + 2y 

4. Required the difference of \by and — r~ • 

118J/-1 

.tj?is. . 

8 

oar GX 

5. Required the difference of r and rTT* 

^ 2axe 
Ans. — ^. 

O^ — o- 

X 

' w 

Zba — 26a; — ex 



G. Required the difference ofx~_ andx-f — . 

Ans. 



2bc 
d 3; a-\-x 

7. Required the difference of a 4 — r- and a— - — -' 

^ a+a; o. — x 

2o24-2a;2 

Ans. — — . 

a^ — x2 

8. Required the difference of ax + — ^ — a"^ a; — 

ffn^ 86x — 99 
21 ' .^Mi. flx f68~~"* 

9. Required the difference of 2xH — "— — , '^'^^ 3x + 

^'-'^— ^Q , 32x4-5 

15 ^'J'^*. ^+-1^^- 

a — X _ 
and 



10. Required the difference of . a + ^. _■ ^ 

j3rts. a- 



a+ic - ^^ 



a(a-x)' « 



2 



•T' 



ALGEBRAIC FRACTIONS. 4t 

CASE VIII. 

To multiply fractional quantities together. 

RULE. 

Multiply the numerators together for a new numerator, 
and the denominators for a new denominator ; and the 
former of these, being placed over the latter, will give the 
product of the fractions, as required*. 

EXAMPLES. 

X 2a; 

1 . It is required to find the product of '- and -^t-- 

Here '-= -— = — the product required. 

6X9 54 27 *■ ^ 

2. It is required to find the continued product of 

_, and 
2' 5 21 

TT a;x4.TXlO.T 403:^ ^.r^ 

Here = = — the product. 

2X5X21 210 21 ^ 

3. It is required to find the product of - and -• 

a a— X 

-^ XX (a+x) x^-\-ax 

Here y (=-7; the product. 

a X (a — X) a'^ — ax 

4. It is required to find the product of— and — r- 

2 3o 



» When the numerator of one of ihe fractions to be maltiplied, and the 
denominator of the other, can be divided by some quantity which is com- 
mon to each of them, the quotients may be used instead of the fractious 
themselves. 

Also, when a fraction is to be multiplied by an integer, it is the same 
thing whether the numerator be miiUipIied by it, or the denominator di- 
vided by it. Or if an integer is to be multiplied by a fraction, or a fraO" 
tion by an integer, the integer may be considered as having unity for ifg 
denominator, and the two be then multiplied together as usual. 

E 2 



42 ALGEBRAIC FRACTIONS. 

6. It is required to find the product of -^ and -_i. 

5a 

6. It is required to find the continued product of — , 
If!, and 4-. , 8a.3 

7. It is required to find the continued product of 

o c ' 26 

8. It is required to find the product of 2a-\- — and 3a — 

_. 26 62 

ax -ins. 6a^-{-3bx — — >. 

a; o 

9. It is required to find the continued product of 3x^ 

2a '*°^a +6' 2a2 4-2a6 

10 It is required to find the continued product of 

flS — a;2 o2 — b' , , ax . o^ — a^^ 

, , and a -{- Jins. . 

«+6 ax+a;2 a — a: x 

CASE IX. 

To divide one fractional quantity by another. 

RVLK. 

Multiply the denominator of the divisor by the nume- 
rator of the dividend, for the numerator ; and the nume- 
rator of the divisor by the denominator of the dividend, 
for the denominator. Or, which is more convenient in 
practice, multiply the dividend by the reciprocal of the 
divisor, and the product will be the q.iotieat required*. 

• When a fraction is to be ilivided by an integer, it is tlie same thing 
•whether the numerator be divided bj it, ov the denominator multiplied 
T)V it. 



ALGEBRAIC FRACTIONS. 43 



EXAMPLES. 

1. It is required to divide ■— by —. 

X 2x X 9 9x- 3 
Here ~^-=.-^X-=-=.^=H .Ans. 

'2a , 4c 

2. It is required to divide -r- "y -j' 

TT ^a d 2ad ad ^ 

Here \/ = = Ans. 

b 4c 46c 26c 

x-\-a x-\-b 

3. It is required to divide ^3-^ by ^^^' 

Here ^±^ ^^Jl J^l±.^Jll±^ Ms. 
X — b x-\-b x^ — 6^ 

4. It is required to divide — rr — 7 by — ; — 

^ a^-\-x^ '' x-\-a 

2a;2 ic+a 2x^(x-\-a) Src 
Here ~ r~, X = 



a3+a;3 x a:(a5+x3) x^ — ax+a^ 

Ix 3 n 7x^ 

5. It is required to divide -~ by — '^"** -|-r* 

4x^ 4x 

6. It is required to divide — — by Cx. Ans.—, 

x-\-l 2x x-4-1 

7. It is required to divide — ~ by tt- '^"^* "TT • 

8. It is required to divide by -=• Ms. - — . 

2ax-\-x^ X 

9. It is required to divide —, — by • 

2a -fx 
^n5. -- 



c- -{-cx-{-x^ ' 

10. It is required to divide _*"'~^_ by fiil^'. 

x2— 26x+62 X— 6 

x2-{-62 



.^71$ 



X 



Also, when ihe two numerators or the two denoininators, cati be di- 
tided by som« common quantity, that quantity ir.av be thrown out of each, 
*nd the quatieats ased instead oi the fractions first proposed. 



44 



INVOLUTION. 
INVOLUTION. 



Involution is the raising of powers from any proposel 
root; or the method of finding the square, cube, biquad- 
rate, &c. of any given quantity. 

RCLE I. 

Multiply the index of the quantity by the index of the 
power to which it is to be raised, and the result will be 
the power required. 

Or multiply the quantity into itself as many times less 
one as is denoted by the index of the power, and the last 
product will be the answer. 

Note. When the sign of the root is ■\-, all the powers of 
it will be + ; and when the sign is — , all the even pow- 
ers will be -f , and the odd powers — : as is evident from 
multiplication *. 

EXAMPLES. 



a, the root. 


a2 the root. 


ft2 =square. 
fl3=cube. 


a*=square. 
a6=cube. 


a*=4th power. 
a* =5th power^ 
&c. 


a8=4th power. 
a» «=5th power. 
&c. 


— 3g the root. 


— 2ax2 the root. 


4-9a2=square. 
— 27a^=cube. 


-}- 4a2 .T« = square. 
— 8a3a;6=cube. 


-f.81a^=4th power. 
&c. 


-j-16o4xs=4th power. 
&c. 



* Any power of the product of two or more quantities is equal to the 
same power of each of the factors multiplied logetber. And any power 
of a fraction is equal m the same power of the numerator divided by the ii 
like power of the denominator. 

Also fl"' raised to ilie nth power is ntm- and -— am raised to the nth 
power is ^ awi, according as n is an ev?u or an odd number. 



INVOLUTION. 



45 



- the root. 
a 

— =»quare. 



a~ 



-=cube. 



-— =4th power. 



'lb 
4a2x4 



the root. 



962 

8a 2. T® 



:square. 



+ ■ 



2763 



:cube. 



816" 



:4lh power. 



X — a the root, 
a- — a 

a; 2 — ax 

x2 — 2ax-\-a~ square. 

X— d 

a;3— 2ax2-fa2x 
— ax2-}-2a2x— a' 

a3_3ax2 4-3a2x-a3 cube. 



.r+a the root. 
x-{-a 

x2-}-ax 



r2 4-2ax4-a" square. 
x-{~a 



x3+2a.r2-fa2x 



;3 4-3ax2-f-3tt2x4-a3 cube. 



EXAMPLES FOR PRACTICE. 

1. Required the cube or third power, of 2a2. 

Arts. 8a«. 

2. Required the biquadrate, or 4th power, of 2a2x. 

Ans. 16a*x*. 

2 

3. Required the cube, or third power, of —^x-y^. 

o 

Ans.- — x^u'-*. 
27 

3a2 X 

4. Required the biquadrate, or 4th power of ■ 



Am. 



662 
"6256" 



46 INVOLUTION. 

5. Required the 4th power of a+.r ; and the 5th power 
of a—?/. Ans. a^-{-4a^x-\-6a~x^-\-4ax^-\-x'^ , and o* 

RULE II. 

' A binomial or residual quantity may also be readily rais- 
ed to any power whatever, as follows : 

1. Find the terms without the coefficients, by observing 
that the index of the first, or leading quantity, begins with 
that of the given power, and decreases continually by 1 , in 
every term to the last ; and that in the following quantity, 
the indices of the terms are 1,2, 3, 4, &c. 

2. To find the coefficients, observe that those of the 
first and last terms are always 1 ; and that the coefficient 
of the second term is the index of the power of the first : 
and for the rest, if the coefficient of any term be multi- 
plied by the index of the leading quantity in it, and the 
product be divided by the number of terms to that place, it 
will give the coefficient of the term next following. 

JVote. The whole number of terms will be one more 
than the index of the given power ; and when both terms 
of the root are +, all the terms of the power will be -J- ; 
but if the second term be —, all the odd terms will be -Ki 
and the even terms — ; or, which is the same thing, the 
terms will be + and — alternately*. 



• The rule here given, which is the same in the case of integral pow 
ers as the binoftiinal theorem of Newton, may be expressed in general 
terms, as follows ; 

'^ 2 '23 



% 

a' 

3, 

)f. 

\ 



{a—b)m=:am—mam-\l,-{-m.—^a'>n-2bs—m.—^ . —:^ am-^b^, &c. 



m — 1 . m — I m — 2 

T"' 3 

which formula? will, also, equally hold when m is a fraction, as will b« 
more fully explained hereafter, 

It may, also, be farther observed, that the sura of the coefficients U 
every power, is equal to the number 2 raised to that power. Thus l-J-i 
=2, for the first power ; l-J- 2.f 1=4=22, for the square; 1-f 3^-3-t 
1=8=23, for the cube, or third power ; and so on. 



f-i, 






EVOLUTION. 



EXAMPLES. 



1. Let a-{-x be involved, or raised to the 5th power. 
Here the terms, without the coefficients, are 

a*, a'^x, a^xz,a^x^, ax*, xs. 
And the coefficients, according to the rule, will be 
5X4 10X3 10X2 5x1 

' ' 2 ' ~3~'~T~' ~1~' 
or 1,5, 10, 10, 5, 1, 

Whence the entire 5tb power of a+.r is 

a^-^5a*x-{-\0a^x2-{-l0a"x3+5ax*+x^. 

2. Let a — x be involved, or raised, to the 6th power. 
Here the terms, without their coefficients, are 

a^, a^x, a*.r2, a^x^, a^x*, ax^ , x^. 

And the coefficients, found as before, are 

6x5 15x4 20x3 15x2 6x1 

' ' -2"' T"' "T"' ~5"' ~6~ ' 
or 1, 6, 15, 20, 15, 6, 1. 
Whence the entire 6th power of a— a; is 

a6-.6a5x+15a*.r2 ^20a^x^-\-\5a^x^ —6ax^-i-xG. 

3. Required the 4thpower of a+a-, and the 5th power 
of a — .T. ^Ins. a* -\-4a^x-{-6a^ x^ -{-4ax^ -\-x''' , and a^ — 

Sa^x+lOa^x^ — lOa^x^-^oax^—xK 

4. Required the 6th power of a+6, and the 7th power 
of a— 2/. Ais. a6+6a56+15a*i2_^20a3^»3_(_i5a254_|_ 

6ai>5+66, and a-?— 7a6i/+2 105 2/2—35 

5. Required the 5th power of 2-\-x, and the cube of 
a-fca;+c. Ans. 324-80a;+80x2+40a;34-10a;*+.T% and 

a3 + 3a2 c+3ac2 -^c^-3^bx — 6acbx — 
3c2 &x-f 3o62 x2 -{- 3co2 ;t2 _ fe 3 ^ 3 . 



EVOLUTION. 

Evolution, or the extraction of roots, is the reverse of 
involution, or the raising powers ; being the method of find- 
ing the square root, cube root, &c. of an}^ given quantity. 



48 EVOLUTION. 

CASE I. 

■ To find any root of a simple qttanttly. 

RULE. 

Extract the root of the coefficient for the numeral part, 
and the root of the quantity subjoined to it for the hteral 
part ; then these, joined together, will be the root re- 
quired. 

And if the quantity proposed be a fraction, its root will 
be found, by talcing the root both of its numerator and de- 
nominator. 

J\i'ote. The square root, the fourth root, or any other 
even root, of an affirmative quantity, may be either -f- or 
— Thus, ^u2=-i-aor — o, and 4/6^=-j-6 or — b, &c. 
But the cube root, or any other odd root, of a quantity, 
will have the same sign as the quantity itself. Thus, 

^ya^=a; l/ — a^ = -a; and V-a5=_a, &e.* 

It may here, also, be farther remarked, that any even 
root of a negative quantity, is unassignable. 

Thus, ^—a^ cannot be determined, as there is no 
quantity, either positive or negative, (-}- or — ), that, 
when multiplied by itself, will produce — a-. 

EXAMPLES. 

1. Find the square root of 9x2 ; and the cube root oi 

Here v' ^^^ = v/9 X ^x^ =.3>:a:=3a; Ans. 
And 3/3a;3 = 3/8X3/x3=2xa:=2a:. Ans. 



* The reason why 4- o and — a are each the square root of 02 i» ob- 
vious, since, b}' the rule of multiplication. {+a) X (-f-a) and ( — a) X 
( — a are both equal to as , 

And for the cube rtot, fifth root, &c. of a neg;ative quantity, it is plain, 
from the same rule, tbat 
(— a)x(— a)X(— a)=— 03 ; and (— a3)X(+a2)=— as. 



Aud consequently i.y — c" ;= — a, and \/ — ao 



•a. 



k. 



EVOLUTION. ' 49 



2. It is required to find the square root of -— - and 

the cube root of , 

2Tc3 

ax , „ . Ba^x^ 2*x 



TT .a~x^ ^a^x2 ax j ,, ^ 

Here */ =z^ -— — ; and 2/ „^. 

4c3 y/4c2 2c ^ 27c3 3c 

3. It is required to find the square root of4a'^x^. 

Jl7is. 2ax^. 

4. It is required to find the cube root of — I25a^x^. 

.fins. — 5ax^. 

5. It is required to find the 4th root of SS^a^ir^. 

Ans. 4ax^. 

6. It is required to find the square root of • 

Jlns. — -. 
3xy 

7. It is required to find the cube root of . 

^ 125x6 

Jlns. — . 
5a;2 

8. It is required to find the 5th root of — ' — . 



Atis 



243 
~2ax^ 



CASE II. 
To extract the square root of a compound quantify. 

RULE. 

1. Range the terms, of which'the quantity is composed, 
...according to the dimensions of some letter in them, begin- 
ning with the highest, and set the root of the first term in 
the quotient. 

2. Subtract the square of the root, thus found from the 
first term, and bring down the two next terms to the re- 
mainder for a dividend. 

F 



60 EVOLUTION. 

3. Divide the dividend, thus found, by double that pari 
of the root already determined, and set the result both in 
the quotient and divisor. 

4, Multiply the divisor, so increased, by the term of 
the root last placed in the quotient, and subtract the pro- 
duct from the dividend ; and so on, as in comrjion arithme- 
tic. 

EXAMPLES. 

1. Extract the square root of a;* — 4a;3+6a;s — 4a;+l; 

a;* —4x3+6x2 —4a;+l(x2 -2a;+l 



X 



2x2— 2x)- 4x3-}- 6x2 
— 4x'-|-4x2 

2x2 _4x-h 1)2x2 -4x+l 
2x2-4x+l 



Ms. x" — 2x-\- 1 , the root required. 
2. Extract the square root of 4a''-|r-12a3x-|-13a?x2-}-6 

.4a« + 12a3x-M3a2x2-}-6ax3+x4(2a2-{-3ax-|-a;2 
4a* 



4a2-}-3ax(l2o3x-{-13a2x2 
12a^a-\-9a2x^ 



4a2 -{-6ax+x2 )4a2 ^^ -\-6ax^+x* 
4a2x2-j-6ax3-f-x* 



* 



Kote. When the quantity to be extracted has no exact 
root, the operation may be carried on as far as is thought 
necessary, or till the regularity of the terms shows the law 
Vy which the series would be continued. 

> EXAMPLE. 

1. It is required to extract the square root of 1+x. 



EVOLUTION. 51 

^^^28 ^16 128' 

1 



^^+i) 



X 






8/4 



a;2 a; 3 .^ji 



x-2 a;3xa;' .x 



64 
8 ''Te 64 ' 256 



a;3 ic* re* a;^ 



a;>^ 



64^64 256 

Kfiere , if the numerators and denominators of the two 
last terms be each multiplied by 3, which will Qot alter 
their values, the root will become 

2 2.4^2.4.6 2.4.6.8 ~2.4.6.8. 10 
where the law of the series is manifest. 



EXAMPLES FOR PRACTICE* 

2. It is required to find the square root of a*-\-4a^x-{~ 
6a^x2-i-4ax3+x*. Ans. a^-{-2ax-{-x^. 

3. It is required to find the square root of x* -2x3-|- 

2 2 ^16 



62 EVOLUTION, 

4. It is required to find the square jroot of 4a;8— 4.r'*4- 
12a;3+a:2-6a;+9. -Ans. 2x^ -x-{-3. 

5. Required the square root of x^ +4x5 -}-lOx'^ +20x^ 
+25.T2+24a;+16. Ans. x^-{'2x^+3x+i, 

6. It is required to extract the square root of a^ 4-6. 

7. It is required to extract the square root of 2, or of^ 
1 + 1. Ans. l+i-i+TV—A+A.^c. 

CASE III. 

To find any root of a compound quantity. 

RULE. 

Find the root of the first term, which place in the quo- 
tient ; and having subtracted its corresponding power from 
that term,l3ring down the second term for a dividend. 

Divide this by twice the part of the root above deter- 
mined, for the square root ; by three times the square of 
it, for the cube root, and so on ; and the quotient will be 
the next term of the root. 

Involve the whole of the root., thus found, to its proper 
power, which subtract from the given quantity, and divide 
the first term of the remainder by the same divisor as be- 
fore ; and proceed in this manner till the whole is finish- 
ed*. 

* As this rule, in higt powers, is often tband to be very laborious, it 
may be proper to obseine. that the roots of various compound quantities 
mav sometimes be easily discovered, as follows : 

Extract the roots of ail the simple terms, and coTinect thcM together by 
the signs + or — , as may be judged most suitable for the purpose \}^^ 
involve the compound root, thus found, to its proper power, and tt it be 
the same with the given quantity, it is the root required. But if it be 
found to difler onlv in some of the signs, change them from + to — , or 
from — to -f-, tili'its power agrees with the given one throughout. 

Thus, in the third example next followi.ig, the root is 2a — 3a;, which 
is the difference of the roots of the fi«t, and last terms } and in the lourth 
example, tiic root is a +6+ c, which is the sum of tiie roots of the tirst, lourOi, 
and sixth terms. The same may also be observed ol th^ sixth example. 
where the root is found from the first and last terms. 



EVOLUTION. 53 



EXAMPLES. 



1 , Required the square root of cf* ~ 2a^x-^3a^ o:^ — 2ax'^ 

4-x*. 

a* — 2a'.T4-3a2 x- — 2ax^-\-x* (a^—ax-^x^ 

a* 



9 



2a~)—^2a^x 



a*— 2a3a;+a2x2 



2a2)2a2a:2 



a* — 2rt3a:+'3a2x2 — 2ax^+x-^ 



2. Required the cube root of x^ + 6.r^— 40x3+96a--»- 
a;6^6x5 -40x3-f-96x— 64(a;3+2a;— 4 



x^ 



3a;<)63 



3x*)'-Ux* 

,;6_}_6a.5_40x3+96.T-64 



3. Required the square root of 4a^ — I2rt;r4-9a:3. 

Ans 2a — 3x. 

4. Required the square root of a~-\-2ab-{■2ac-{■h^-\- 
2bc-\-c^. Ans. fl+i+c. 

5. Required the cube toot of x^—Gx^^lox*—2Qx^-{' 
16a:2 _6a-|-l. Ans a;— 1. 

6. Required the 4th root of ISa* -96a3a:+ 216x2 x^-. 
216as:3+81a-^ Ans. 2fl— 3x. 

F 2 



bi IRRATIONAL QUANTITIES, or SURDS. 

7. Required the 6th root of 32x^—QQx'^-\-80x^ — 40x''t 
4-IOj:— 1. Ans. 2x— 1. 



Of irrational QUANTITIES, 
OR SURDS. 



Irrational quantities, or surds, are such as have no 
exact root, being usually expressed by means of the radi- 
cal sign, or by fractional indices ; in which latter case, the 
numerator shows the power the quantity is to be raised to, 
and the denominator its root. , 

Thus, ^2, or 2 = , denotes the square root of 2 ; and 

3/a3, or a3, is the square of the cube root of a, &c. * 

CASE I. 

To reduce a rational quantity to the form of a surd. 

RULE. 

Raise the quantity to a power corresponding with that 
denoted by the index of the snrd ; and over this new quan- 
tity place the radical sign, or proper index, and it will be 
of the form required. 

EXAMPLES. 

1. Let 3 be reduced to the form of the square root. 
Here 3X3=32=9 ; whence v^9 Ans. 



* A quantity of the kind here mentioned, as for instance ^ 2, is called an 
irrational number, or a surd, because no number, either wliolc or fractional, 
can be found, which when multiplied by itself, will produce 2. But its ap- 
proximate value may be determined to any degrree of exactness, by the com- 
mon rule for extracting the square root, teing 1 and certain noo periodic de- 
tisQals, which uever terminate. 



"7 



IRRATIONAL (QUANTITIES, or SURDS. bb 

2. Reduce 2x^ to the form of the cube root. ^ 

Here(2x2)3=8x« ; whence 3/8x6, or (8.x«)'-i' 

3. Let 5 be reduced to the form of the square root. 

Ans. ^{25) 

4. Let— 3a; be reduced to the form of the cube root. 

Ans. 3/- (27x3). 

5. Let — 2a be reduced to the form of the fourth root. 

Ans. -y(16a4). 

6. Let a2 be reduced to the form of the fifth root, and 

a/«+i/^- ^^and to the form of the square root. 

Ans. V «' ^ x/ (a+2^ab-\-b), v/ (i «). and y/ li- 

Note Any rational quantity may be reduced by the 

above rule, to the form of the surd to which it is joined, 

and their product be then placed under the same index, or 



radical sign. 



EXAMPLES. 



Thus 2^2=^4X^^2=^4X2=^8 
And 23/4 = 3/8 X 3/4= y 8X4 = 3/32 

Also 3ya=v/9Xya=v'9Xa=y9a 

And ^3/4a=3/i-X3/4a=3/yX4a=3/i 
J. Let 5^6 be reduced to a simple radical form. 

Ans. yd 50). 

2. Let ^'v/5a be reduced to a simple radical form. 

Ans y(|) 

2a 9 

3. Let — l/— — be reduced to a simple radical form. 

3 4tt2 ^ 



CASE II. 



, 2a 
Ans. %/-Z' 

t3 



To reduce quantities of diff'erent indices^ to others that 
shall have a given index. 

RULE. 

Divide the indices of the proposed quantities by the 



b6 IRRATIONAL qUANTITlES. or SURDS. 

given index, and the quotients will be the new indices for 
those quantities. 

Then, over the said quantities, with theirnew indices, 
place the given index, and they will be the equivalent 
quantities required. 

EXAMPLES. 

i i- 

1. Reduce 3^ and 2^ to quantities that shall have the in- 
dex i. 

1116 6 , , . , 

Here ;:-f-77=xX-=-=3, the 1st index : 

2 6 2 12 

And _^_=-X-=-=2, the 2d index. 

Whence (33)6 and (22)6, q^ 276 and 46,are the quan- 
tities required. 

J. 1 

2. Reduce 5^ and 6^ to quantities that shall have the 

1 J- -. 

common index-. Ans. 1256 and 366* 

6 

3. Reduce 22 and 4* to quantities that shall have the 

1 i 1, 

common index - Ans. 16^ and 16'" 

o 

4. Reduce a- and a^ to quantities that shall have the 
common index - Ans. o\ tnd . \ 



5. Reduce a- and 6^ to quantities that shall have the 
1 

6* 



1 1 _ 

common index -. Ans. (a^)* and {b*y. 



Note. Surds may also be brought to a common index, 
by reducing the indices of the quantities to a common de- 
nominator, and then involving each of them to the power 
denoted by its numerator. 



IRRATIONAL QUANTITIES, or SURDS, 57 



EXAMPLES. 
I 



1. Reduce 3^ and 4^ to quantities having a common m- 
dex. 

Here 32 = 3«=<^' '6=21)' 
And 43=4«'=42*)^=r6)^ 

Whence 2l\'^ and 16]^. Ans 

2. Reduce 4^ and 6* to quantities that shall have a com- 
mon index. 

Ans. 256^5 and 125^2. 

3. Reduee a^ and d^ to quantities that shall have a com- 
mon index. 

Ans. aA^ and a^je. 
± _i 

4. Reduce a"-* and b* to quantities that shall have a com- 
mon index. 

Ans. aT\h andfe^''2. 
J. I 

5. Reduce a" and 6"' to quantities that shall have a com- 
mon index. 

Ans. a^ml and b"lnn. 

CASE III. 
To reduce surds to their tn-ost simple forms. 

RULE. 

Resolve the given number, or quantity, into two factors, 
one of which shall be the greatest power contained in it, 
and set the root of this power before the remaining part, 
with the proper radical sign between them.* 



» When the given surd contains no factor that is an exact power of the 
kind required, it is aheadv in its niosl simple form. 

Tlius, v' 15 cannot be reduced lower, because neither of its factors, 5, nor 
3;^ is a square, 



68 IRRATIONAL QUANTITIES, or SURDS. 

EXAMPLES. 

1. Let ^48 be reduced to its most simple form. 

Here ^48=^16 X3=4<^3 Ans. 

2. Let ^108 be reduced to its most simple form. 

Here y 108=3/27x4=33/4 Ans. 

Aoie 1. When any number, or quantity, is prefixed tc 

the surd, that quantity must be multiplied by the root o 

the factor above mentioned, and the product be then joinec 

to the other part, as before. 

EXAMPLES. 

U Let 2^32 be reduced to its most simple form. 

Here 2^32—2^16X2=8^2 Ans. 
2. Let 5^24 be reduced to its most simple form. 

Here63/24=53/83<3=103/3 Ans. 
Note 2. A fractional surd may also be reduced to a mor€ 
convenient form, by multiplying both the numerator an^ 
denominator by such a number, or quantity, as will make th« 
denominator a complete power of the kind required ; ant 
then joining its root, with 1 put over it, as a numerator 
to the other part of the surd.* 

EXAMPLES. 

2 
1. Let v^- be reduced to its most simple form. 




the 

question <i^i/,c gi>cu, ivjitre 11 is louiiu iiiai 'V^^^'^ V ^^'i "' vvuicu case u 
^ oniy necessary to extract the square root of the whole number 14, (or to 
find it in some of the tables that have been calculated for this purpose) and 
then divide it by 7 ; wliereas, othenvise, we must have first divided the nu- 
merator by the denominator, and then have found the root of the quotient, for 
the surd {lart; or else have deterrgined the root both of the numerator and 
denonjinator, and then divided the one by the other; which are each o^ them 
troublesome processes when performed by the common rules ; and in the nex( 
example, for the cube root, ie labour would be much greater 



7 



IRRATIONAL QUANTITIES, or SURDS. 59 

ere •?=vii=v'(^X14)=iv't4 Ans. 

2. Let 3^- be reduced to its most simple form. 
"'' =V^=3Vy||=3y(4x50)=|v50 A„s. 

EXAMPLES FOR PRACTICE. 

3. Let v/126 be reduced to its most simple form. 

Ans. 5^5. 

4. Let v'294 be reduced to its most simple form. 

Ans. 7^6. 

5. Let 1/56 be reduced to its most simple form. _ 

Ans. 23/7, 

6. Let ^192 be reduced to its most simple form. 

Ans. 4^3. 

7. Let 7^80 be reduced to its most simple form. _ 

Ans. 28.^5. 

8. Let 93/81 be reduced to its most simple form. 

Ans. 27^3. 

3 5 
;}. Let- — ^- be reduced to its mostsimjple form. 

Ans. ^1^ v^30. 

10. Let-^ — be reduced to its most simple form. - 

Ans. 1^12. 

1 1 . Let ^9da-x be reduced to its most simple form. 

Ans. la^lx. 

12. Let.yz3_a2,x2 be reduced to its most simple form, 

Ans. .TV's; — a^. 

CASE IV. 

To add surd quantities together. 

RULE. 

When the surds are of the same kind, reduce them to 



60 IRRATIONAL QUANTITIES, or, SURDS. 

their simplest forms, as in the last case ; then, if the sure 
part be the same in them all, annex it to the sum of the 
rational parts, and it will give the whole sum required. 

But if the quantities have different indices, or the sure 
part be not the same in each of them, they can only bf 
added together by the signs -\- and — . 



EXAMPLES. 

1. it is required to find the sura of ^27 and ^48. 

Here ^21=^ 9X3 =3^3 
And ^48=^16X3=4^/3 

Whence 7^3 the sum. 

2. It is required to find the sum of 3/500 and l/WS. 

Here 3^500=3/1 25X4 =53/4 
And 3/ 108 =3/ "27X4 =3 3/4 

Whence 8^4 the sum. 

3. It is required to find the sum of 4^/147 and 
v/^o. 

Here 4^147=4^ ^4.9 x 3 =2>y3 

And 3^ 75=3^25X3=15^3 

Whence 43^3 the sum. 

2 1 

4. It is required to find the sum of 3^- and 2^ — 

c. 2 „ 10 3 ijtjf 

Here3^-=3y-=-^10 ;; 

Jiff, 



And 2v'— =2^ =— v'lO 

^10 ^100 10^ 



4 
Whence ~x/lO the sum 



4t 
1. 



IRRATIONAL qUANTITlES, or SURDS. Gi 



EXAMPLES FOR PRACTICE. 

6. It is required to find the sum of ^72 and ^^128. 

Ans. 14v'(2). 

G. It is required to find the sum of ^180 and ^^405. 

' Ans. 15^(5). 

7. it is required to find the sum of 35/40 and ^135. 

Ais. 93/(5). 

8. It is required to find the sum of 4^54 and 53/128. 

Ans. 323/(2). 

9- It is required to find the sum of 9^^243 and 10^363. 

Ans. 19V(3). 

. 2 27 

to. It is required to find the sum of 3^- and 7*/ — . 

Ans. 3J^^(6). 

Jl. It is required to find the sura of 123/- and 33/_ 

4 32' 

Ans. 6^3/(2). 
12. It is required to find the sum of | ^a'^b and 

^y46.xv. Ans. (^+^)^6. 



CASE V. 

To find the difference of surd quantities. 

RULE. 

When the surds are of the same kind, prepare the qnim- 
tities as in the last rule ; then the difference of the rational 
parts annexed to the common surd, will give the whole 
difference required. 

But if the quantities have different indices, or the surd 
part be not the same in each of them, they can only be 
subtracted by means of the sign — . 

1. It is required to find the difference of v'448 and 
v/112. 



G 



62 IRRATIONAL QUANTITIES, or SURDS. 



Here ^^448 =^ 64X7 =8^7 
And v'112=^16X7=4y7 

Whence 4^7 the difference. 

2. It is required to find the difference of P/192 and- 
3/24. 

Here yi92=3/64'x3=43/3 
And 3/24 =3/8X3 =2^/3 

Whence 23/3 the difference. 

3. It is required to find the difference of 5*/20 and 
3^/45. 

Here 5^20=5^4X5=10^5 

And 3^45=3^9X5= 9^/5 



Whence ^b the difference. 

4^3' 



3 2 
4. It is required to find the difference of-^-, apd 



2 1 
6^i- 



TT 32363^1^ 
2 12 6 2 1 



Whence ^-v/6 the difference. 



60 
ortinswer required. 



EXAMPLES FOR PRACTICE. 

1. It is required to find the difference of 2^50 and 
^1-8. I p 

Ans. 7^(2). I 5 



IRRATIONAL QUANTITIES, or SURDS. 63 

2. It is required to find the difference of y 320 and 
3/40. ♦ Ans. 23/(5). 

3 5 

3. It is required to find the difference of ^-^nA^/ — . 

Ans. 4*5^/(1^)- 

4. It is required to find the difference of 2^/^ and ^Z. 

Ans. v/(2)- 

5. It is required to find the difference of 3 \/\ and y72. 

Ans. V(9)- 
2 9 

6. It is required to find the difference of \/- and V'^' 

Ans. tVVC^S). 

7. It is required to find the difference of V SOa^x and 
^20o2 x3. Ans. (4o2 — 2ax)y(6x). 

8. It is required to find the difference of 8 {/a 3 6 jyid 
2ya«6. Ans. (8a-2a2)3/(&). 

„Vofe. The two last answers may be written thus, 
(2ax— 4a2)^(6x), and 
(2aa-8a)i/(t). 



CASE VI. 

To multiply surd quantities together. 

RULE. 

When the surds are of the same kind, find the product 
of the rational parts, and the product of the surds, and 
the two joined together, with their common radical sign 
between them, will give the whole product required ; 
which may be reduced to its most simple form by Case iii. 

But if the surds are of different kinds, they must be re- 
duced to a common index, aod then multiplied together as 
usual 

It is also to be observed, as before mentioned, that the 
product of different powers, or roots, of the same quanti- 
ty, ig found by adding their indices. 



€4 IRRATIONAL QUANTITIES, or SURDB. 



\ 



EXAMPLES. 



J. It is required to find the product of 3^8 and 2^/6. 
Here Sy/S 

Multiplied 2v'6 



Gives 6 v/48=6-v/ 16x3=24^3 Ans. 

12 3 5 
S. It is required to find the product of -^- and -%/-^' 

Here ~l/- 

31ultiplied -3/r 
^ 4*^ 6 

3 10 3 5 3 15 

1-1 
3. It IS required to find the product of 2^ and 3^. 

Here 2^=2^= (23)^=8^ 

And 33=36= (32 )^=9« 



Whence (72) i Ans. 
4. It is required to find the product of 5\/a and 3^a. 

J. ^ 
Here 5\/a=5a2=5a« 

And 33/a=3a3=3a8 



Whence 15a6 = l5(a^)« or IS^a^ Ans. 



EXAMPLES FOR PRACTICE. 

S). it is required to find the product of 5^/8 and 3^5. 

Ans. 30^(10). 
6. It is required to find the product of '^18 and 5^4. 

Ans. 10^(9). 



IRRATIONAL OTJANTITIES, ok SURDS. 66 

1 2 ■* 

7. Required the product of -,/6 aad— v/9. 

Ans- tWC^)- 

8. Required the product of -v^l8 and 5^20. 

Ans. 15y(10). 

9. Required the product of 2^3 and 13i ^5 ^ 

Ans' 'i7^(15)^ji«^ 

10. Required the product of 72ia.^ and 120ia*. 

Ans. 87061a H- 

11. Required the product of 4-f2^'i and 2-^2. 

An?. 4. 
± X 

!2. Required the product of (a+fc)" and (a+i)'"- 

Ans. (a +6) M n . 

CASE VII. 

7'o divide one surd quantity by another. 

RULE. 

When the surds are of the same kind, find the quotient 
of the rational parts, and the quotient of the surd.f. and the 
two joined together, with tlieir common radical sign be- 
tween them, will give the whole quotient required 

But if the surds are of different kinds, they must be 
reduced to a common index, and then he divided aa before. 

It is also to be observed, that the quotient of different 
powers or roots of the same quantity, is found by sub- 
tracting their indices. 

EXAMPLES. 

1. It is required to divide 8 -v/ 108 by SyS. 

G 2 



G6 IRRATIONAL QUANTITIES, or SURDS 

2. It is required to divide 83/512 by 4 ^2. 
Here -^^—21/256=21/64X4=81/4 Ang. 

3. Itis required to divide-^5 by-^2. 

i^5 3 5 3 10 3 ' ^ ^ 
Here f5!l^=-^-=-^_=-^10 Ans. 
iy2 2^2 2^ 4 4^ 

4. It is required to divide ^7 by ^/l. 

*/7 72 7« 3-2 i 

Here ^=1:— =— =78 "=76 Ans. 
V ♦ 73 7I 

5. It is required to divide 6^64 by 3^2. 

Ans. 6^3. 

6. It is required to divide 43/72 by 2 y 18. 

Ans. 23/4. 

• , , •, .3 1,21 

7. It IS required to diviae ^-:\/-r^T"y-^'\/r- 

Ans. 11^3. 

5 2 2 3 

8. It is required to divide 4;:^'^ by 2 -Vj- 

Ans. 14^*2. 

1 2 

9. It is required to divide 4- ^ a by 2- %/ab. 

27, a, 4 

2 3 

10. It is required to divide 32 -^ a by 13-^a. 

648 A 

3 - 9 JL 

11. It is required to divide 9-a"by4— a»n. 

825 »"-" 
424 

12. It is required to divide ^20+^ 12 by v'd+^S. 

Ans. ^4. 

Note, Since the division of surds is performed by sub- 



IRRATIONAL QUANTITIES, or SURDS. 6? 

tracting their indices, it is evident that the denominator of 
any fraction may be taken into the numerator, or the nu- 
merator into the denominator, by changing the sign of its 
index. 



o 
Also, since — = 1, or = a'""'"=a», it follows, that the 

expression a" i-s a symbol equivalent to unity, and conse- 
quently, that it may always be replaced by 1 whenever it 
occurs.* 

EXAMPLES. 

_, 1 a-i ,1 a" 

i. Thus -= — , or o"^ : and — = — , or a". 
a \ a" 1 ' 

^ ., b bu-2 , ^ a-« 1 b"' 

2. Also, -r=-7-, or 6a 2 ; and r^=— 7-„,or — . 

a^ I 6'"* a"6-"» o" 

3. Let — be expressed with a negative index. 

Ans. a-2. 

4. Leta-^ be expressed with a positive index. 

* Ans. -^. 
ai 

5. Let — I — be expressed with a negative index. 

a-\-x 

Ans. (a-fa:)->. 

6. Leta(a2— a;2)-J be expressed with a positive index. 

Ans 



a{a'~x^')x' 



* To what is above said, we may also farther observe, 

1. That added to or subtracted from any quantity, makes it neithergreat- 
er nor less ; that is, 

a~\-0=a, and a— 0=a. 

2. Also, if nought be multiplied or divided by any quantity, both the pro- 
•duct and quotient will be nought ; because any number of times 0, or any 

, part of 0, is ; that is, 

OXa,or oXOrrO, and - =0. 

a 

3. From this it likewise follows, that nought divided by nought, is a finite 
^antity, of some kind or other. 



68 IRRATIONAL QUANTITIES, or SURDS. 

CASE VIII. 

To involve, or raise surd quantities to any power. 

RULE. 

When the surd is a simple quantity, multiply its indeii 
by 2 for the square, by i for the cube, &c and it will give 
the power of the surd part, which being annexed to the 
proper power of the rational part, will give the whole 
power required. And if it be a compound quantity, mul- 
tiply it by itself the proper number of times, according to 
the usual rule* 






For since OX «=0i or 0=:0 ^a, it is evident, that — -^a. 

4. Farther, if any finite quantity be divided by the quotient will be ia- 
finite. 

Forlet-=.?, then, if 6 remains the same, it is plain, the less o is, the 
a 

arreater will be^the quotient q ; whence, if a be indefinitely small, g will be 
indefinitely great : and consequently, when a is 0, the quotient q will be in- 
finite : that is, 

Which properties are of frequent occurrence in some of the higher parts of 
the science,' and should be carefully remembered. 

Since, therefore, "XT is the same as (a+b) ~ ' . Let us suppose, in the 

general formula, n .::— 1; and we shall have for the coefficients n= — 1: 



=. — 1 ; —^ 1 ; —7--"= — 1, &c. and for the powers 



3 ' 4 



oj a we 



71 -1 1 n—\ —2 1 n— 2 1 71—3 1 . 
have a =.a r:r-i« =« — — ;« = — ;a = — &c. 

a — a2 a 3 a 4 

-) 1 1 b , ij2 b^ b4 b5 
so that (a 46) = ;= f- — ——Ti > &c. which i: 

the same series that is found by division. For more on this subject see th« h 
Binomial Theorem, (further on) or Eulers Al'^ebra. '■ 

* When aav quantity that is affected with the sign of the square root is 1< '*lll 
be raised to the second power, or sciuared, it is done by suppressing the sign 
Thus. ' 

y/a)2 , or \^'aX\/a=a; aad (v/a-J-/))3, or v^fl-f-6X\/a+6=a-f-ft 



Ifi 



JID 



IRRATIONAL QUANTITIES, or SURDS. 69 



EXAMPLES. 



2 J- 

1. It is required to find the square of ~a^. 

/S i\ 4 1X2 4 f 4 
Here(-a»)==^a" =-^a' ^^V-'- Ans. 

2 

2. It is required to find tjie cube of ^-v/S. 

Here^X3*=|^V'27=lv^93^=|v'3 Ans. 

3. It is required to find the square of 5^3. 

Ans. 9^9. 

4. It is required to find the cube of 17v''21. 

Ans. 103173^(21). 

6. It is required to find the 4th power of - -v/ 6. 

Ans. -j'g. 

6. it is required to find the square of 3+2 v/ 5- 

Ans. 29+12^5. 

7. It is required to find the cube of ')/x-\-3\^y. 

- Ans. x^x-{-21yy/x-{-9xx/y+21y^y. 

8. It is required to find the 4th power of ^3 — ^2. 

Ans. 49-20^/6. 



CASE IX. 

To find, the roots of surd quarJities. 

RULE. 

When the surd is a simple quantity, multiply its index 
y i for the square root, by i for the cube root, k.c and 
: will give the root of the surd part ; which bemg; annex- 
d to the root of the rational part, will give the whole roev 



70 IRRATIONAL QUANTITIES, or SURDS. 

required. And if it be a compound quantity, find it« root 
by the usual rule.* 



EXAMPLES. 



1. It is required to find the square root of 9^3. 
Here (9y3)2=92 x. 33^2=9^X36=36/3 Ans. 

2. It is required to find the cube root of r-v/^. 

o 

Here(iy2)^=(^^^^X(2^^^)=i(2«)=iV'2 Ans, 

3. It is required to find the square root of 10'. 

Ans. 10^(10). 

Q 

4. M is required to find the cube root of — -a<. 

Ans. fa^a. 

16 3. 

5. It is required to find the 4th root of —-as. 

81 

Ans. |a^. 

a a 
Q. It is required to find the cube root of ^^/e* 

Ans. ^|, or ^-i^/iSa). 

7. It is required to find the square root of a;^ —Ax^i 
-|-4a. Ans. x— 2^a. 

8. It is required to find the square root of a-}-2y/a6+i 

Ans. ^a-\-^h. 



* The nth root of the mth power of any number o, or the mtb power c 

, , . . m, 

the nth root of a, is a—. 
n 

Also, the nth root of the mth root of any number a, or the with root of th 
1 

nth root of a, IS a 

From which last expression, it appears that, that the square root of th 
square root of a is the 4th root of a ; and that the cube root of the square iw 
of a, or the square root of the cube root of a, is the 6th root of a ; and so o 
/or the fourth, fifth, or any ather numerical root of this kind. 



IRRATIONAL QUANTITIES, or SURDS. 7't 



CASE X. 



To transform a binomial, or a residual surd, into 
a general surd. 

RULE. 

Involve the given binomial, or residual, to a power cor- 
responding with that denoted by the surd ; the» set the ra- 
dical sign of the same root over it, and it will be the ge- 
neral surd required. 

EXAMPLES.- 

1. It is required to reduce 2+ v/ 3 to a general surd. 
Here {2 +^sy = 4+3-f 4 y/ 3 = l-\-i^3 ; therefore 

2-\-^3=y/l-{-4\/3, the answer. 

2. It is required to reduce y'S-j-y'S to a general surd. 

Here (y2-f ,- v/3)- =2+ 3+2^/6 = 6+2 V 6 ; 
therefore V 2+^3= V 6+2^6, the answer. 

3. It is required to reduce 3/2+^4 to a general surd. 
Here(^/2+V4)3=6+63/2+63/4; therefore 3/2+ 

3/4=3/6(1 +v/2+v/4), the answer. 

4. It is required to reduce 3— ^5 to a general surd. 

Ans. ^(14-6^5). 
3. It is required to reduce ^2— 2^6 to a general surd. 

Ans. V(26-4v'12). 

6. It is required to reduce 4— \/l to a general surd. 

Ans. v'(23-8a/7). 

7. It is required to reduce 2^/3 - 3^-9 to a general surd. 

Ans. C/(1623,9-108;>3-219). 



72 IRRATIONAL QUANTITIES, or SURDS. 

CASE XI. 

To extract the square root of a binomial, or residual surd. 

I 

RULE.* 

Substitute the numbers, or parts, of which the given 
surd is composed, in the place of the letters, in one of the 
two following formulae, according as it is a binoaiial or a 
residual, and it will give the root required. 

And if the second part of the binominal, or residual, in 
this case, be an imaginary surd, the same theorems will 
still hold, by only changing —b into -f-6, as below 
^{a+^-b)=^{^a+^^{a^+b))+^{^a~.l^{a^^b)) 

* Prop. 1. From the lemma, pagte 16. vol. ii. Bonnycastle's Algebra; it 
appears, that if tivo numbers a and b be prime to each other, the fraction 

— can never be a whole number. 
(iin 

Prop. 2. From the same principles it likewise follows, that no whole num- 
ber can have a vulgar fraction for its root. 

Prop. 3. It may, also, be farther proved, that the square, cube, &c. root 
of a whole numb»r, cannqj be partly rational and partly irrational. 

For, if possible, let, in the first place, \/a=x+^i/, for the square root. 

Then, by squaring both sides, a==X2 +2x^y-\~y ; and by transpositiMi, 
^^2Tv/^=a — X2 —y. 

(^ ^2 — y 

Whence, by division, ±\^yssi = a rational quantity; which 

is contrary to what has been proved in the last proposition. 

Prop. 4. In any equation of die form 3"+: y'y=av^ +6, the rational parts, 
on the opposite sides, are equal to Each other, and also their irrational parts. 

For, if X be not equal to a, lot it be equal to a+^. Then aHr.ziv'y^ 
a+ vf' '> o'" -•- v'^'^ — ^:il v'y- 'fhat is, the ^b is partly rational and 
partly irrational, vvhich, by the last proposition, has been shown to be impos- 
:-ible. 

Whence, x=^a, and consequently ^'y=^b. 

Prop. 5. It may here also be shown, that the squaite root of a binomial 
or residual surd, may sometimes be made equal to the sum, or difference of a 
whole number and a surd, or of two surds. Thus, let \/(a + \/6)=^x-J^j/ ; 



IRRATIONAL Q,UANTrf lES, or SURDS. 73 

Where it is to be observed, that the only cases that are 
useful in this extraction, are when a is rational, and a^ — 6 
in the first of these formulee, or a^-\-b in the latter, is a 
complete square. 

EXAMPLES. 

1. It is required to find the square root of l\-\-^l9,-ov 

Here, 



and 



Whence v/(11+6v'~)='^+a/2' *^^ answer required. 
2. It is required to find the square root of 3—2^2. 

Here, 



v/JQ +i-y/a" - 6=x/ | + ^a/9-- 8^^/t+l=^/2 ; and 

= -1; 

Whence v'(3 — 2^2)=^2— 1, the answer required. 

3. It is required to find the square root of 6±2y/ 5. 

Ans. ^5±:i. 

4. It is required to find the square root of 23±8^7. 

Ans, 4±^7. 



jfcen, \/ (a — y/ h) =x — y. And, by squaring each of these equations, 
xve shall have a -f" \/ it = 3:2 •\-2xy + y^ , and a — ^6 = xs — 23cy 

. 2 2 2 2 

J_y2 ; whence, by adaition, 2a=2x •+2y , or a=x -{-y . 
And, by multiplying the like sides of the same two equations, there will aj-ise 

2 9 '^ 2 

(tt't~v/^)^(« — y/b)=x — y , or ^/(aa — b)=x'' — y . 
Wherefore, by adding and subtracting, 

" + ^^ (''2 — 6)=2ia , and a — v' (a2— 6)=2y2 ; or 
2-"=\/(2«+iV (a2—b)), and y=^{-Ka—L^(a2—b')). Whence, 
v/(«+ Vi)=V(i«4-|'^(«2— 6))+ v'aa— 5v^(a2— ft)), and 
^(a— ^b)^,/Hn + 14/(0.2— b)) — s/(\a—i>/(a2—b) . 
And, if 6 be negative instead of affirmative, the two formula, in that case, 
will become s/ {<^-i- ■/—b)=->/ (^(1 + 1^-/ {a2 + h))4-'/-{ia—% ^ (as+b)) 
and ^{a—y —l)—^(^a-f- 4 v/ (as ~\-b))—^(\—l-^ht3'+b)} ; hence 
■♦he above rule is evident. Q. E. D. 

H 



U IRRATIONAL (QUANTITIES, or SURD3. 

5. It is required to find the square root of 36 ±. 10 
y/ 11. Ans. 5 ± ^ (11). 

(^ It is required to find the square root of 33 ± 12^6. 
t Ans. 3 ± 2 y6. 

7. It is required to find the square root of 1 + 4 y/— 3, 
or 1 -f v^ — 48. Ans. 2 + ^—3. 

8. It is required to find the square root of 3±4^— I, 
or3±:y'-16. Ans.2±^—\. 

9. It is required to find the square root of — 1+\/ — 8. 

Ans. 1 + ^-2. 

10. It is required to find the square root of a^-f- 2 x^/ 
(a2 — x2). Ads. x + ^ {a x — x"). 

11. It is required to find the square root of 6 + 2 ^'' 2 
- ^ (12) - V (24). Ans. 1 + ^2 — v/3. , 

For Trisomial, Qcadrin'omial Scrds, &c. 

Rule. Divide half the product of any two radicals by 
a third, gives the square of one radical part of the root ; 
this repeated with different quantities, will give the squares 
of all the parts of the root, to be connected by + aud — . 
But if any quantity occur oftener than once, it must be 
taken but once. 

For if iH-y-r? be any trinomial surd, its square will be 
x2 J_ y2 _^ 22 4_ Q-xy X 2xz + 2]/2 ; thfitt if half tbc pro- 
duct of any two rectangles as 2xj/X2xz (or 2x2 j,r) be di- 

2x2 wz 
vided by some third ^yz, the quotient ■ = x- , must 

needs be the square of one of the parts ; and the like 
for the rest. 

EXAMPLE 1 . 

To extract the square root of \0 + y/ (24) + ^ (40'; 
+ V'vc;o). 

V (24) X y/ (40) _ ^ (24) X ^ (6 0) 

""^" iVT^O) ^' '"^ 2 v/ (40) 

y9=3,aBd^^^A|L^|^=^(25) = 5. And the 
rootis v/2+-v/3-f v/5. 



-tF 



IRRATIONAL QUANTITIES, or SURDS. 75 

EXAMPLE 2. 

It is required to find the square root of 12 + \/(32) — 

V(48) + >/(80) -y(24) + v/(40) — v/(60). 

Here ^^^ --^^ = '^-^, this prodeces nothing. 
^ . v/(32X48) ,,,^. , , , y(4Q X 60) 

, V V(32X40) ,, „ ^(48X24) 

= ^(25)=5 ; and^^^-^ =^4=2 ; and -^-^^^ 

■= V 9 = 3 ; and ^^'7^^ = ^<'^^ = '' ^'* '^'''" 
fore the parts of the root are ^4, ^^5, ^3, ^^, ^/'i, kc. 
and the root 2 + ^2 - ^"3 + ^6 ; for being squared 
It produces the surd quantity given. 

CASE XII. 

To extract any root (c) of a binomial surd. 

RULE I*. 

Let the quantity be A ± B, whereof A is the greater 
part and c the exponent of the root required. Seek the 

' w ■ — ■ ' 

* Let the sum or difference of hco quantities x ani y be raised to apoieer 
whose exponent is c, and let the Ist, 3d, 5th, Ith, cfc. terns of that power ^ 
collected into one sum, be called A, and the rest of the terr>is, in the even 
olaces, call B ; the difference of the squares of A and B shall be equal tolhe 
difference of the snuares of x and y raised to the same power c. 

for the tenus in the c power of a;-4- y writing for their coefficients, res- 
c c — 1 c — ^2 c— 3 

pectively, l,c, rf, c, &c. area; -f c^ y -^ d'x y2 -^-p.x y3-f&c. 
=^A •{• B, and the sanie power of x — y (changing- the signs in the even places) 

c c — 1 c — 2 c — 3 

s X — c a; y -\. dx y2 — e x J^ 3 -f- &c. = A — B, and thei-efore 

> + y) X (^— y) =iA + B)X (A-B]=A.2 ~~B2 =.)(xiy}X ( r-v) I 

c 
=(xs—y2) . Q. E. D. 



76 IRRATIONAL QUANTITIES, or ^SURDS. 

least numbern whose power «' is divisible by A^ — B^ , the 

quotitint being Q, com|>ute ^(A-f-BX^Q) in the nearest 
integer number, which suppose to be r. Divide A ^ Q, 
or by its greatest divisor, ;xnd let the quotient be a, and let 

71 

?• -f- r 

— - — = t, the nearest integer. Then the root = 

t s± v/(<2s2— ») 



V^ 



if the c root of A±B can be extracted. 



Let one, or both of the quantities a', y, be a quadratic surd, that is, let 
'• -j-3/' *'^^ ^ root of the proposed binomial A~j~ £ belong to one of these 
forms, p -}■ i y/ fj, k \/ p ■+■ q, or k^/p -\- l\^q- And it follows : 1. If 
,, -{-)/ =p -^l^fj, c being any whole number. A, the sum of the odd terms, 
A\ ill be a rational number ; and B, the sum of the terms in the even places, 
each of which involves an odd power of y, will be a rational number multi- 
plied into the quadratic surd ^/g. 

2. Let c, the exponent of the root sought, be an odd number, as we may 
ahva} s suppjse it, because if it is even, it may be halved by the extraction of 
the square root, till it becomes odd ; and let x -^ y = k ^ p +9. Then A 
^vill involve the surd v P, and £ will be rational. 

3. But if both members of the root are irrational, (x-i-y=kv p + l\/q) 
A and B are both irrational, the one involving \^p, and the othe* the surd 
\/q. And in all these cases, it is easily seen that when x is greater than y, 
A will be greater than B. From this composition of the binomial A-i~Bj 
•ve are led £0 its resolution, as in the foregoing rule, by these steps. 

I. 

When A is I'ational, and A2 — B2 is a peri'ect c power. 

1. By tiie theorera,A2 — B2 =(a-2— y2)3 accurately; and therefore ev 
tracting the c i-oot of A 2 — Ba it will be x'-i —ij2 , call this root n. 

2. Extract in the nearest integer the c ro9t of A-i-B, it will be {nearly) 
■X "f- y, which put = r . 

3. D;-.ade x2 — y2 (=n) t>y x-^-y (=■ r) the quotient is (nearly) x—y; 
and the sum of the divisor and quotient is (more nearly) 2x ; that is, if an 

iateger value of a: is to be found, it will be the nearest to T- 

2 2 2 2 ^r-j.n^2 2 

4. X —{x —y )—y ■■,,0T, f C ) —n = y -. whence y — 

^'•+!lx2 r-^n 
^(f ^ ) — n), and therefore putting i = T, the root sought = t 

\ 2 y 2 - 

+ \/ ('2 — n) ; the same expression as in the rule, when Q = 1, s = 1, thai ., 
IS when As — B2 is a perfect c power, and the greater member A is ra-l ^ 
tional. 



IRRATIOKAL QUANTITIES, or SURDS. 77 



EXAMPLE. 




What is the cube root of ^ 968 -{- 23. 

We haveAa — B^ = 343= 7X7 X7. Q , X7=» = n^ 

whence «= 7, and Q,= 1. Then y(A + B X ■>/ Q,) = 
3/56 + = r = 4. A ^ Q. = y 968 == 22 v/ 2, and the 

radical part y'S = s, and ^ o c 

2^ i 



. II. 

■When A is irrational, and Q= 1. By the same process, at= ^ 

2 
(= T) andy ■= v'(T3 — w). But seeing A is supposed irrational, and c 
an odd number, x will be irrational likewise ; and thty will both involve the 
same irreducible surd ■^p, or s, which is found by dividing A by its greatest 
rational divisor. Write, therefore for x or T, its value t X s, and x -{-y ,— . 
<s + \^{t2 s2 —n). 

III. 

If the c root of As — B2 cannot be taken, multiply A 2 — B2 by a num- 

berQ, such that the product niav be the {least) perfect c power »''(:= A2 Q 

— B2 Q.) And now (instead of A-f-B) extract thee root of (.\ 5-B)Xv/Q, 

which found as above, will be < s -}- y/ {t2 ns — n) ; and consequently the 

- c root of A -f B will be ^ s -j- y/{t2. &3 — n), divided bv the c root of v' Q • 

that is, ''+^^''Jr--£L. 

In the operation, it is required to find a number Q, such, that {\2 — B2) 
X Q may be a perfect c power ; this will be the case, if Q be taken equal 
fo (A 2 — B2)t— I ; but to find a less number which will answer this condi- 
tion, let A2 — B2 be divisible by a, a, (wi) ; 6, b, . . . . (n) \ d, 

■m » r 
d,...,(r);&c. in succession, that is, let A 2 — B2 z= a b d &c. also, 

X y z in \. X n-\-y '■ + ^ 

IetQ = a6 rf &c. (A2 — B2) X Q=a X^ ^ <^ ^^• 

which is a perfect cth power, if a, y, z, &c. be so assiuiied that w-f-^s '*+y> 
r-f-z, are respectively equal toe, or some multiple of c. Thus to find a 
number which multiplied by WO will produce a pcrtect cube, divide 180 as 
often as possible by 2, 3, 5, &c. mid it appears that 2. 2. 3. 3. 5 = 180 ; \i, 

3 3 iJ o 

therefore, it be multiplied by 2 . 3 . 5 . 5, it hiecomes 2 . 3 . 5, or (2 . 3 . 5) ; a 
perfect cube. 

If A and B be divided by their greatest common measure, either integer 
or qiiadrutic surd, in all cases wh<;re the cth root can be obtained by diis 
aiethod, Q will either be unity, or some power of 2, less than 2'. 

If the residual A— B be givtn, it is evident from its genesis by involution , 

H g 



78 IRRATIONAL QUANTITIES, oh SURDS. 

nearest integer. And <s=2^2, ^/{i^ s^ — «) = ^{Z -Tj 
= 1. 6/(Q=l. And the root is -^-^— = 2 V 2 
+ 1, whose cube, upon trial, I find to be ^968 + 25. 

RULE n.* 

Let the surd, that is to have its root extracted, be of 
the form ^/{a -j- v^ *), or ^/(a — ^ b). Then if a^ —b 



thftt the same rule gives its root x — y. See Universal Arithmetic, p. 139. 
Dr. Waring's Med. Alg. p 287, or Maclaui-in's Alg. p. 124. 
n 

* Thus, let^(a + v' t) = * + V y i aid we shall have, by involution, I 
n ' 

An equation which, by expanding the right hand member, and comparing 
the rational and irrational parts, gives 

« . n(»— 1) »*— 2 , n (w— 1) («— 2) (w— 3) «-^ 2 
"=^H 2 — * ^"^ 2.3.4 y +&C. 

»»— 1 . n (n— 1) (/I— 2) ^—3 . . „ 

V'6 = na: v'y + — ^ — 273 ' " y^y+ ■■•+&<^ 

Or which is the same thing, under a different form, 

, t rt n> 

, t n n> 

vfc = 2 M^+v'y)^(^-^y) ) 

Whence by squaring each of these equations and subtracting the latter 
(rom the former, we shall have 

2 4 2re 2 » 2n ) 

a-b=-^}{x+ ^y) +2(a;-y) + («- v'y) $ 
, ( 2«. 2 71 2n} , 

-Jli^-h^/y) -2(ap-y) + (af-v/y) 5 
Or, by rejecting the terms that destroy each, and then multiplying by v, 
2 2 w 2 2 j^. 

a — b= (x — y) , or a; — y= (a — 6)H" 
2 
Where supposing a — 6 to be a complete power of the nth degree, let 
2 i 2 "2 

(a — b)n be put = c. Then since x — y = c, and consequently y=x — c, 

« w(n — 1) "~2 
if this value be substituted for y, in the equation x + — — jj — - x y-J. 

w(n— 1) (« — 2U« — 3) « — 42 v „ . '• 
■ - - ■ ' X y + &C' ■="1 we shall obtain an equa- 
tion in which the value of x, as before mentioned, is rational, when the e^ 
iracticm requirad is possible. 



IRRATIONAL QtJANTITIES, or SURDS. 79 

ke a perfect integral cube, and some whole number, can 
be found, that, when substituted for n, will make 

?i3 _ 3 (3/a2 _ 6) n = 2 a, 
the roots of the two expressions, in this case will be 

^(a + v' ^) = i« ^ i \/("^. - 'I v«EE3 

y{a - ^ 6) = In - 1 v/ (n^ - 4 ^a^ - 6) 
And if the second part of the binomial, or residual, be an 
maginary surd, and a^ -|- ^ be a perfect integral cube, the 
sxtraction may be etfected, by finding the integral value 
>f n in the following equation as before. 
n3 — 3 (3/a2 + h) n = 2a. 
In which laist case, the roots of the two expressions 
irill be, 

;ach of which formulae may be obtained, by barely chang- 
ng the sign of b in the former. 

i EXAMPLE. 

It is required to find the cube root of 10 ± 6 ^ 3, or 

I0±y(108). 

Here a = 10, and b = 108 ; whence ^(a2 _ fe) = 3^ 
>00 — 108) = - 2, and n^ — 3 (s/^i^j n — 20, 

or n3 -{- 6n = 20 
vhere it readily appears, from inspection, that n = 2. 
Vhence ^(10 + v/108) = t + i ^(4 - 4 X - 2> = 

+ i -v/02) = 1 + v^3, and i/(lO- V 108) = f - J 
/(4 - 4 X - 2) = 1 - ^ V 12 = 1 _ ^3. 

EXAMPLES FOR PRACTICE. 

). Required the cube root of 68 — .^4374. 

Ans. 



2. Required the cube root of 11 + 5 ^1. 



Ans.^/^+' 



i/2 

V3 



80 IRRATIONAL QUANTITIES, or SURDS. 
3. Required the cube root of 2 y/l -\- 3 ^ 3. 



Ans. 



2 



4. Required the tifth root of 29 ^6 + 41 ^/3 

Ans. ^^±^. 

6. Required the cube root of 45 ± 29 ^2. 

Ans 3 -i- ^2, and 3 — ^2. 

6. Requi/ed the cube root of 9 ± 4 ^b, or 9 ± ^SO. 

Ans 1 + 1 ^5, and | - | ^ —5. 

7. Required the cube root of 20 ± 68 ^ —7. 

Ans. 5 + ^ — 7, and 6 — ^ — 1. 

8. It is required to find ;he cube root of 35 ±69 ^ — P. 

An« 6 + ^ — 6. and 5 — V^— 6. 

9. It is required to find the cube root of 81 ■±. ^ ~ 
2700*. Ans. -3+2^ -3, and —3 — 2^-3. 



* Whenever it can be done, the operation, in cases of this kl -d, ought to bej 
abridged, by dividing the given binomial by the gres'^ st cube that itcontainst 
and then finding the root of the quotient ; which being multiplied by the root oj; 
the cube, by which the binomial was divided, will give the root required i 



Thus iu the example above given, 81 -f v'— 2700 = 27 X (3 + ^- 



2 7 M 



I u 



where the root^of 34- v' — 2 7 > beingnow more easily found to be — 1+ 2^ 

— -5, — 1 + 3 v* — 3, we shall have by multiplying by 3, (which is the cub 
root of 27), — 3 -{- 2 v — 3, as above. 

Also, this is useful, in Cardaus' rule for cubic equations ; thus, •v/(81 + y* 

—3 

|_2700)) -f- v/(81— v/ (— 2700))=— 3x 2 = — 6, or = ~ X 2 = — 3 

or ^ X 2 = 9, the imaginary parts vanishing, by the contrariety of their sign 
See'De Moivre's appendix to Sanderson's Algebra, Universal Arithnietio, o 
Maclaurin's Al-j^bra. 



IRRATIONAL QUANTITIES, or SURDS. 81 



CASE Xlll. 



To find such a multiplier, or multipliers, as will make any 
binomial surd rational. 



RULE.* 

1. When one or both ^f the terms are any even roots, 
multiply the given biorainai, or risidual, by the same ex- 



*^The demonstration of this rule is evident from the following theorems. 

THEOREM I. 

mm n — m n — 2m m n — 2m2irL 
Generally, if you multiply a — h by a -f- a 6 + a b 

n — 47ft 2m ^ 

4" a b , &c. continued till tlie terms be in number equal to ^, the 

n n fix — m n—Hm m n — 3/rt 2?« n — 4?w 

pK>duct«hallbeo — b : for,( a +« b +a b +« 

3 m n — m\ m m 
h &c. b J X(a —b ). 



n 


n — m n 


n 


-2 m 2 


m 


n- 


-3 


m '3 m 


a + 


a b + 


a 


b 


+ 


a 




b 


&c. 


n 


— m m n — 2 


m 2m 


n~ 


-Am 


3m 


a 


— a 


b +a 




b - 


-a 




b 


&c. 


—b 



jj * ^ * fF <F jj 

a —b 

THEOREM II. 

n — m n — 2m m n — Sot 2m n—4m 3m la m, 

a — a 6 -{- a 6 — a b &c. multiplied by (a + 1) 

\ n n n 

gives a -f 6 , which is demonstrated as the other. Here the sign of b is 

n 
positive when m is an odd number. When any binomial surd is proposed, 
suppose the index of each number equal to m, and let n be the least integer 

n — m n — 2 m m n — 3 m 
number that is measured by m ,- then «hall a ^ a b -i-a 

2m 

h &c. give a compound surd, which multiplied into the proposed surd 
m m 
a "^ b , will give a rational product. Thus to find the surd which ^nulti- 

plied by 2/ a — \/ b, will give a rational quantity. Here m,= ^, and th^ 



S2 IRRATIONAL QUANTITIES, or SURDS. 

pression, with the sign of one of its terms changed ; and 
repeat the operation in the same way, as long as there are 
?urds, when the last result will be rational. 



n — in 
least number which is measured by ^, is unit ; let n. = 1, then shall a -f- 

n — 2m m n — 3 to %m 1 — i. 1 — 2 j. o 3 2 i j. 

o b -t a b .&c.=a =^+a ^ b^ -\- a b^ --=a^ + aH^ 

+6^=V (a2) ^'\/(ab)+ \/(62),which multiplied by V a— V6,£ives 
a — b. 

THEOREM III. 

m I n — TO n — 2 m I n — 3 m (21) 

Let a ^6 be multiplied by a Hpa 6-f-a ^^ 

n — 4 TO {31) n M HV 

a b -f- &c. and the product shall give o -t-fcw : therefore w must be 

taken the least integer that shall give m. also an integer. 

n — m n — 2 to I n — 3 to (2/)_ n — 4 to (3Z) o » 

Dem. a ZiZa b -f.* * ■+■« ^ i &c. a 6)«— l^Z 

TO I 

X_(a ±h). ^ 

n n — TO / n — 2 to (2/) 
a -f-a b "^a b &c. 

n — TO I n — 2 TO (20 ^l 

itfi b —a b &c. — 6i7t < 

« * * » -J- 1^ 

a ,m . 

■n_l Vl 

The sign of bm is positive only when m is an odd number, and the binomial 
TO I 

proposed is a -|- 6 . 

If any binomial surd is proposed whose two numbers have different indices, 
let these be to and I, and n equal to the least integer number that is measured 

m n — TO n — 2m T n — 3 to (2/) n — 3 me (3/) 

by TO and by 2 ; and a -|-o b -j-a b 4^a b &.c. 

TO I 

shall give a compound surd, which multiplied by the proposed a — 6 , shall 
give a rational product. 

1 wi 

Thus %/a — %/b being given, suppose m = %l=s 3^, and T=f, therefore 

n—m n—2m I n—3m {21) n—4m{3l) 3—^ 

n = 3,anda +a 6 +« ^ -f- a b •f&c.=a " 

3—1 J. 3—3 2 14^ 

-fa 63^a -b'^+ab-i.a'^b'^-{-b^ = \/(a5) + a2-}i\/b + 

y/{a^) Xy/b2 ^ab +\/a Xy/b* + %/bS=a2 y/a ^azy.\/b ^ 

«\/ «X\/ fc2 4-06-^6 v/aX V^' + ^v' *«> "'hich mulliplied b;-- 

n "' 
\/«— V (*)i gives o — b m=a 3—62 . 



IRRATIONAL QUANTITIES, or SURDS. 8? 

2. When the terms of the biominal surd are odd roots, 
the rule becomes more complicated ; but for the sum or 
difference of two cube roots, which is one of the most 
useful cases, the multiplier will be a trinomial surd, con- 
sisting of the squares of the two given terms and their 
product, with its sign changed. 



EXAMPLES. 

1. To find a multiplier that shall render 5+ -v^S ra- 
tional. 

Given surd 5 4- -\/3 
Multiplier 5 — ^3 

Product 25 — 3=22, as required. 

2. To find a multipher that shall make ^/o-{- ^3 ra- 
tional. 

Given surd ^5+^3 
Multiplier .^5-^3 , 

Product 5 — 3=2, as required. 

3. To find multipliers that shall make yS + V^ ^^' 
iional. 

Given surd {/b+^/3 ~- 
J st multiplier y5— 4/3 

1st product ^5 — ^3 
2d multipher ^5+^3 



2d product 5_3=2, as required. 

4. To find a multiplier that shall make \/l-\- 1/3 ra- 
ional. 



By these Theorems any binomial stJrd whatsoever being; given, you may 
nd a surd, which multiplied by it shall ^ive a rational prcduct See Mae- 
wriJi's Algebra, page 112. 



84 IRRATIONAL QUANTITIES, or SURDS. 

Given surd 3/7-f- 3/3 

Multiplier ^^1^—1/(7 X3)+^S^ 



7+ 3//3X73) 
— y(3X72)«3/(7X32) 

+ ^(7X32)+3 

Product 74-3 = 10, as was required. 

5. To find a multiplier that shall make ^ 5 — ^x ra- 
tional. Ans. y/5-\-^x. 

6. To find a multiplier that shall make \/ n -\- ■)/ b ra- 
tional. Ans ^a — ^h. 

7. To find multipliers that shall make a-\- tj b ra-| 
tional. Ans. a— y/h. 

8. It is required to find a multiplier that shall makej 
1 - 3/2a rational. Ans. l+^2a+^4a2. 

9. It is required to find a multiplier that shall mal 
y3-iV2 rational. Ans. y9+X3/6+i3/4. 

10. It is required to find a multiplier that shall makt 
y(a3) -f- V(63), or a| + 6| rational. 

Ans. ya9r-y(a9^2) ^ yfas^e^ _ 4^59. 



CASE XIV, 



To reduce a fraction, whose denominator is either a simplt ,, 
or a compound surd, to another that shaU have a rationc ' 
denominator. 

RULE. 



1. When any simple fraction is of the form — , mu 
tiply each of its terms bv ^a, and the resulting fracticlHere 
will be ^. 



lU 



IRRATIONAL qUANTlTlES, or SURDS. 86 

Or when it is of the form ;; — , multiply them by ^o^, 

and the result will be -^ — . 

a 

And for the general form - — , multiply by ya"-i, and 

the result will be -^ . 

a 

2, If it be a compound surd, "find such a multiplier, by 

(he last rule, as will make the denominator rational ; and 

multiply both the numerator and denominator by it, and 

the result will be the fraction required. 



EXAMPLES. 

2 3 
1. Reduce the fractions — and , to others that 

shall have rational denominators. 

Here -i.=^,X^J=!^ ; and-^=^X^ = 
v/3 v^s v/3 3 ' ^y5 xy5 %/h^ 

. 54/63 64/53 6 , ' . 

r = — - — -='ts/\~o the answer required. 

. Reduce—— — -- to a fraction, whose denominator 
shall be rational. 

\ Here— i— x'^^y::^=^-y:^^=!^^V2__ 

v/3-^2 V/5+V2 5-2 3 '^ 

■= V'ij+v^S the answer required. 

•v/2 
3. Reduce to a fraction, whose denominator 

o— Y^ i 

shall be rational. 

Here-^ - v/2X (3+^/2) _3y2+_2_2-f 3^2 
'3-v'2 (3- v' 2) X (3+^2) 9—2 T~" 

2 3 
~7"^7 v^2 the answer required. 



86 IRRATIONAL QUANTITIES, or SURDS. 

4. Reduce — ;-— to a fraction, that shall have a ra- 

tional denominator. 

4 

X 

5. Reduce —-. to a fraction that shall have a ration- 

al denominator. 

. Sx — x^x 
Ans. — - — - — . 

9—x 

tl /h 

6. Reduce — ~-r to a fraction, the denominator of 

which shall be rational. Ans. ^r—- 

a^—b 

7. Rediice — to a fraction that shall have Si ra- 

1/1-%/b 

tional denominator. 

Ans. 5X(^(49)+V(35) + \/(25)). , 
3/3 

8. Reduce — — ^- — - to a fraction that shall have a ra- 

3/9+3/10 

tional denominator. 

33/9+33 /(10)4-3/( 360) 
Ans. - . 

4 

9. Reduce . — • — to a fraction that shall have a ra- 

tlenal denominator. 

Ans, -v/(10)-V2+(2— v'5)XV5. 



ARITHMETICAL PROPORTION, &c. 87 



OF 



AKlTHMETiCAL PROPORTION 

AND PROGRESSION. 

Arithmetical Proportion, is the relation which two 
quantities of the same kind, have to two others, when the 
difference of the first pair is equal to that of the second. 

Hence, three quantities are said to be in arithmetical 
proportion, when the difference of the first and second is 
equal to the difference of the second and third. 

Thus, 2, 4, 6, and a, a-f-6, a-{-2b, are quantities in arith- 
metical proportion. 

And four quantities are said to be in arithmetical propor- 
tion, when the difference of the first and second is equal 
to the difference of the third and fourth. 

Thus, 3, 7, 12, 16, and a, a-f-6, c, c-^-b, are quantities 
in arithmetical proportion. 

ARiTHft'ETicAL PROGRESSION is when B scries of quan- 
tities increase or decrease by the same common differ- 
ence. 

Thus, 1,3, 5, 7, 9, &c. and a, a+d, a-{-2d, a-\-3d, kc. 
are increasing series in arithmetical progression, the com- 
mon differences of which are 2 and d. 

And 16, 12, 9, 6, kc. and a, a—d, a— 2d, a— 3d, kc. are 
decreasing series in arithmetical progression, the common 
differences of which are 3 and d. 

The most useful properties of arithmetical proportion 
and progression are contained in the following theorems : 

1. If four quantities are in arithmetical proportion, the 
pum of the two extremes will be equal to the sum of the 
two means. 

Thus, if the proportionals be 2, 6, 7, 10, or a, b, c, d ; 
then will 2+10=5+ 7, and a-f-^=5-}-c. 

2. And if three quantities be in arithmetical propor- 



88 ARITHMETICAL PROPORTION 

tion, the sum of the two extremes will be double the 
meaD. 

Thus, if the proportionals be 3, 6, 9, or a, b, c, then 
will 3+9=2X6 = 12, anda+c=26. 

3. Hence an arithmetical mean between any two quan- 
tities is equal to half the sum of those quantities. 

Thus, an arithmetical mean between 2 and 4 is =— tt— 



~3 ; and between 6 and 6 it is = — ;;— =51. 



6+6 
Y 



And an arithmetical mean between a and 6 is .* 

4. In any continued arithmetical progression, the sum 
of the two extremes is equal to the sum of any two terms 
that are equally distant from them, or to double the mid- 
dle term, when the number of terms is odd. 

Thus, if (he series be 2, 4, 6, 8, 10, then will 2+ 10= 
4 + 8=2X6.-=12. 

And, if the series be a, a-\-d, a-\-2d, a-^Sd, «+4«?, 
then will a + (a+4ci)=(a+cf) + (a+3fZ;=2X(a+2J.) 

5. The last term of any increasing arithmetical series 
is equal to the first term plus the product of the common 
difference by the number of terms less one ; and if the 
series be decreasing, it will be equal to the first term rni- 
nus that product. 

Thus, the nth term of the series a, a-\-d, n+2J, a-{-3d, 
« + 4(i, &-C. is a + (?i — 1)(/. 



* If tH'o, or more, arithmetical means between any two quantities be pe^ 

tjuired, they may be expressed as below. 

_, 2n 4-6 ,o4-26 .... , j i I 

Thus, — 5— and — - — = two aritiimetical means betw'een a and 0, a| 

being the less extreme and b the greater. 

And — ——, r-^ , -= — ■ , &c. to — — -i^anv number (n)\ 

€.f arithmetical means between a and b ; where — ;— is the common differ- 

n-f-1 

ence; which beinj^ added to a, g^ives t)ie first of these means; and hei 

s^jfiin to this last, give; the second ; and so on. 



1 



AND PROGRESSION. 89 

And the nth term of the series a, a—d, a — 2d, a— St/, 
a-4d, &c. is a— (ri— l)d. 

6. The sum of any series of quantities in arithmetical 
progression is equal to the sum of the two extremes n-iul- 
tiplied by half the number of terms. 

Thus, the sum of 2, 4, 6, 8, 10, 12, is = (2 + 12) X 

£=14X3=42. 
2 

And if the series be a-{-{a+d)-\-{a-\-2d)-^{a+3d)+ 
(a+4rf) &c. . . . +/, and its sum be denoted by S, we 

shall have S={a-{-l)X-^, where / is the last term, and n^ 

the number of terms. 

Or, the sum of any increasing arithmetical series may 
be found, without considering the last term, by adding the 
product of the common difference by the number of terms 
less one to twice the first term, and then multiplying the 
result by half the number of terms. 

And, if the series be decreasing, its sum will be found 
by subtracting the above product from t\vice the first tcrm,^ 
and then multipljing the result by half the number of 

terms, as before. s , / , cj\ 

Thus, if tlie series be a+(a+^)+(a+2c/)+(.7+orf) 

j^ (^a + 4d), &c. continued to n terms, we shall have 

S= ^2a-{-{n-l)dl X-. 

And if the series be a+{a-d) + {a—2d)-[-{a-^3d)+ 
U—4d), &c. to n terms, we shall have 

S = J2a-(n-l)cZ| X^(*)- 



r*1 The sum of any number of terms (n) of the seiies of natural numbers 

l,2,3,4,'6,6,7,&c.is = — 2 

■ . . 100X101 „ 
Thus, 1 +24.3+4+5, &c. continued to 100 tenns, is = ^ — =iKJ 

101 X =5050. 

3 2 



90 ARITHJVIETICAL PROPORTION, 4-e. 



EXAMPLES. 



1. The first term of an increasing arithmetical series is 

3, the common diiference 2, and the number of terms 20 ; 
required the sum of the series. 

First, 3+2(20- 1)=3+2X19=3+38=41, the last 
term. 

20 20 

And (3-1-41) X— =44 X— =44 X 10=440, the sum re- ^ 

quired. 

. on 

Or, f2X3+(20-l)X2}x^ = (6+19X2)X10=(6 

+38)X 10=44X10=440, as before. 

2. The first term of a decreasing arithmetical series is 
3:00, the common difference 3, and the number of terms 
34 ; required the sura of the series. 

First, 100-3(34— 1)=100-3X33=100-99=1, the 

last term. 

34 34 

And (1004-1)X-- = 101X— =101 X17= 1717, the 

sum required. 

34 
Or, {2X100-(34— 1)X3]X— =(200-33X3)X17 

=(200- 9g)X 17 = 101X17=1717, as before. 

3. Required the sum of the natural numbers, 1, 2, 3, 

4, 5, 6, &c. continued to 1000 terms. Ans. 600500. 

4. Required the sum of the odd n^imbers 1, 3, 5, 7, 9j 
tc. continued to 101 terms. ^bs. 10201. 



A!s«> the sum of any number of terms (n} of the series of odd numbers 
1 3 6 7 9 11 &c is =: «2. 
' Thus, i -f 3 + 5+7 4-9, &c. continued to 50 terms, is = 502 _ 2500. 

And if any three of the quantities a, d, n, S, be given, the fourth may be 
Jgund from the equation ^ 

S=52«±(«-l)djxJor(« + Zjx| 

"Where the uppK>r sign -f- is to be used when the series is increasing, and the 
lower sign— when it if ^creasing; also the last {erm/=iO+ {n-~\)d^^% 



GEOMETRICAL PROPORTION. 91 

5. How many strokes do the clocks of Venice, which 
go on to 24 o'clock, strike in a dn^ ? Ans. 300. 

6. Required the 365th term of the series of even num- 
bers 2, 4, 6, 8, 10, 12, &c. Ans. 730. 

7. The first term of a decreasing arithmetical series is 

10, the common difference-, and the number of tecvns 

o 

21; required the sum of the series. Ans. 140. 

8. One hundred stones being placed on the ground, in 
a straight line, at the distance of a yard from each other ; 
how far will a person travel, who shall bring them one by 
one, to a basket, placed at the distance of a yard from the 
iirst stone ? Ans. 5 miles and 1300 yards. 



OF 

. GEOMETRICAL PROPORTION 

AND 

PROGRESSION. 

^ GEOMF.TniCAL PROPORTION, is the relation which two 
quantities of the same kind have to two others, when the 



* If there be taken any four proportionals, a, 6, c, d, which it has been 
usual to express by means of points : thus, 

a : I : : c : d, 

ff c 

this relation will be denoted by the equation — = -, ; where the equal ratios 

d 
are represented by fractions, the numerators of which arc the antecedents, 
and the denominators the consequents. Hence, if each of tlie tivo numbers 
«if this equation be multiplied by bd, there will arise ad z=zbc. From which 
it appears, as in the common rule, that the product of the two extremes of 
any four proportionals is equal to that of the means. And if the third c, in 
this case, be the same as the second, or c =6, the proportion is said to be 
continued, and we have ad =6 2, or, b-—^ad\ where it is e\ident, that the 
product of the extremes of three proportionals is equal to the square of the 
mean : or, that the mean is equal to the square root of the product of thct wa 
extremes. 



92 GEOMETRICAL PROPORTION 

antecedents, or leaain,ter.s of each pair are the .a., 
parts of their consequents, or the consequeiu 



cedents. 



I 



A,„ if e«h »™l,ev of .he .equ.tio, «4 =6c be s»cccs*cly divided by 

^ a: C : :t: a 
proportions i 

\b:a::d:C 

&c. 



a b . 

~c'~'d 
b d 



plication, the following equivalent forms : — = ^ ; ^ rf- 

" • '•'," ■ ■ - " *- (^_ „.,(<;no- fhp term l; 



.nt sides of the same eq^iation, we have ■^-^,,. ^r.P 
ent sides on ,n . .m . . c'" • (^"'. In which cases m and 

in the form of a P'-^P^'-^'^P^^^V^n.be^' whatever. 
n'rrSrebTtS^^-everal equations. 



-=-, ' which correspond i 

b d 1 

•^ Z the proportions \ ■ . j^ . . i , ^ 

^ '^^ &C. 



weshallhave,bymultiplyingthe«.liketerms,^;^^y^^^<,. ''^il^i'^'j 
. . tin^ the expression into the form of a proportion, aet &c. . 6 
Or, by pmmg the e:.p a c ^3 ^fore, we shall have, 

&c.::c^i&c.:to&c.. Also, taking -^ =-^, asbetor , 

. «« '"' ■ and by augmenting or diminishing each side J. 
mult^hcauon, ^ ^^-^,^ -^^ ^^nj, ^^«_l,.^eh,- " 

theequationbyl;^J±l-„d^l'^'' . J^_^^ "^6 : -.mc+Ti ,. 
^S expres^d in^efonnof apro^rtron.g.ves».± 
nd;ovma±nb.mc±ncl..n ^ ^ v,e put by a similar «DV »: 

And ?f tbe above mentioned e(riation^=-^, be put y 



i 



AND PROGRESSION. 93 

And if two quantities only are to be compared together, 
the part or parts, which the antecedent is of its consequent, 

plicationof its terms, under the form —7-= -^7 and then augmented or dinii- 
■^ qb qd 

nished by 1, as in the last case, there will arise pa+qb : pc + qd : : qb : qd. 

Whence, dividing each of the antecedents of these two analogies by their 

.. , ... , ma-t-nb nb b .pa-^qb qb 

eonseqwents, the result will g:ve = — r:= — r^=^— ,i and — — ^ , =r — r 

mc-jr_rM nd d pc+:_qd qd 

=— . And, consequently, as the two right hand members of these expres- 

, 6 , „, ma-^nb pa + qb 

sions are each =—., we shall have - — = — -■=. ~. 

a ?nc+:»itt pc + qd 

Or, b\' converting the corresponding teiTns of this equation into a propor- 
tion tna-^nb : mc + nd : : pa^+^qb : pc-jl-qd. Also, because tlie common 

a c a b ^ , . 

equation — = -j gives — =--, if the latter be put under tlie equivalent forme 

ma 7nb pa pd . „ u* • u • -i 

— =r — -, and — = -r, we shall obtain, by a sunuar process, mo-f-nc : 

nc nd qc qd -^ '^ ' — 

/)a+ qc ■.■.mb + nd:pb + qd\ which two analogies may be considered as 
general formulfe for changing the terms of the proportion a : b : -. c : d, with- 
out altering its nature. Thus, by supposing m, n, p, q, to be each=l, and 
taking the antecedents with the superior signs, and the consequents with the 
inferior, we have a-f- 6 : a—b : : c-f- rf : c — d, and a-\-c : a—c : -. b•^•d -. 
I — d ; which forms, together with several of those already given, are the 
usual transformations of the common analog}' pointed out above. 

In like manner, by taking m, n und p each =T, and 9=0, there will arise 
a-hb:a::c-+-d:c, and a+c:a::6 + rf:6; each of which proportions 
may be verified by making the product of tlie extremes equal to that of the 
means, and observing that ad=-bc. 

Lastly, taking any number of equations of the form before used, for ex- 
a c e ff 
pressing proportions, as — c=— =-7.^^— :=:&c. -, which, according to die 

common method are called a series of equal ratios, and are usually denoted 
by a : b : : c : d :: e :/■■■■ g ■• h :: &c. we shall necesssarily have from the 

fractions being all equal to each other --=: 9, ^=9, ^=?, "T=?> ^'^■ 

And by multiplying q by each of the denominators, a^=bq, cr=dq, e=ifq, 
g=hq, &c. 

^Vhence, equating the sum of all the terms on the left hand side of these 
equations, with those on the right, we have a -f c -f e^g -f- &.c.={b-^d-\- 
f^h -f &c.)q. And consequently, by division, and the properties of propor- 
tionals before shown. 

g-l-c-fe-f-g-f&c. __ « __ °4g_"-f c ^ e__ ^ 
b -f- d-f/-f A + &c. ~~ 6 b-^d" b+d+f ~ 

which results show, that, in a series of ecjual ratios, the sum of any number 
of Oie antecedents is to that of their consequents, as one, or more of the an- 
tecedents, is to one, or the same number of consequents, Q. E. D. 



94 GEOMETRICAL PROPORTION 

or the consequent of the antecedent, is called the ratio ; 
observing, in both cases, always to follow the same method. 

Hence, three quantities are said to be in geometrical 
proportion, when the first is the same part, or multiple, 
of the second, as the second is of the third. 

Thus, 3, 6, 12, and a, ar, ar'^ , are quantities in geome- 
trical proportion. 

And four quantities are said to be in geometrical propor- 
tion, when the first is the same part, or multiple, of the 
second, as the third is of the fourth. 

Thus, 2, 8, 3, 12, and a, ar, b.^ br, are geometrical pro- 
portionals. 

Direct proportion, is when the same relation subsists 
betwen the first of four terms and the second, as between 
the third and fourth. 

Thus, 3, 6, 6, 10, and a, ar, b, br, are in direct propor- 
tion. 

Inverse, or reciprocal proportion, is when the first and 
second of four quantities are directly proportional to the 
reciprocals of the third and fourth : 

Thus, 2, 6, 9, 3, and a, ar, br, b, are inversely propor- 
tional ; because 2, 6, -, -, and a, ar, -r-, - are directly 

y 3 or b 

proportional. 

Geometrical Progression is when a series of quan- 
tities have the same constant ratio ; or which increase, or 
ecrease, by a common muliplier, or divisor. 

Thus, 2, 4, 8, 16, 32, 64, kc. and a, ar, ar'' , ar,^, ar* , 
&c. are series in geometrical progression. 

The most useful properties of geometrical proportion 
and progression are contained in the following theorems : 

1. If three quantities be in geometrical proportion, the 
product of the two extremes will be equal to the square 
of the mean. 

Thus if the proportionals be 2, 4, B, or a, b, c, then 
will 2X8=42, and aXc—b^. 

2. Hence, a geometrical mean proportional, between 
any two quantities, is equal to the square root of their 
product. 



AND PROGRESSION. 96 

Thus, a geometric mean between 4 and 9 is =^36=6. 
And a geometric mean between a and 6 is = ^ab*. 

3. If four quantities be in geometrical proportion, the 
product of the two extremes will be equal to that of the 
means. 

Thus, if the proportionals be 2, 4, 6, 12, or a, h^c, d; 
then will 2X12=4X6, and a Xc?=:6Xc. 

4. Hence, the product of the means of four propor- 
tional quantities, divided by either of the extremes, will 
give the other extreme ; and the product of the extremes, 
divided by either of the means, will give the other mean. 

Thus, if the proportionals be 3, 9, 5, 15, or a, b, c, d; 

,, ... 9X5 , ,3X15 ^ , 6Xc - , 

then will =15, and =9 : also, = a, and 

3 o a 

aXd 



c 

5. Also, if any two products be equal to each other, 
either of the terms of one of them, will be to either of the 
terms of the other, as the remaining term of the last is to 
the remaining term of the first. 

Thus, if ad— he, or 2X 15=6X5, then will any of the 
following forms of these quantities be proportional : 

Directly, a : 5 : : c : v-Z, or 2 : 6 :: 5 : 15. 
Invertedly, 6 : a :: rf : c, or 6 : 2 : : 15 : 5. 
Alternately, a : c : : 6 : (Z, or 2 : 5 :: 6 : 15. 
Conjunctly, a : a-\-b '.'. c : c-\-d, or 2 : 8 : : 5 : 20. 



* If two, or more, geometrical means between any two quantities be re- 
quired, they may be expressed as below : 

^asb and ^ab2 =:two geometrical means between a and b. 
i/a 3 6, \/a2 63 and ^ab 3 = three geometrical means between a and h. 

And generally, 
1 1 1 

'a%f"^^ (a^~ '62)""*"^, (a"~*63)""^^~ ^'^^' ""'"^®'" (") °^ geome- 

tncal means between a and 6. Where (_')Ji +1 is the ratio : so that if a 

be multiplied by this, it will give the first of these means ; and th4« last being 
again multiplied by the same, will give the second ; and so on. 



96 GEO]\IETRICAL PROPORTION 

Disjunctly, a : b^a :'. c : d^c, or 2 : 4 : : 5 : 10. 
Mixedly, b-\~a -.b^a :: d+c : d^c, or 8 : 4 :: 20 : 10. ■ 

In all of which cases, the product of the two extremes ia . 
equal to that of the two means. 

6 In any coatinue.d geometrical series, the product of 
the two extremes is equal to the product of any two means 
that are equally distant from them ; or to the square of 
the mean, when the number of terms is odd. 

Thus, if the series be 2, 4, 8, 16, 32 ; then will 
2X32=4X16=82 

7. In any geometrical series, the last term is equal to 
the product arising from multiplying the first term by 
such a power of the ratio as is denoted by the number of 
terms less one. 

Thus, in the series 2, 6, 18, 54, 162, we shall have 
2X3''=2X81 = 162. 

And in the series a, ar, ar^ , ar^, ar'^ , kc. continued to n 
terms, the last term will be 

8. The sura of any series of quantities in geometrical 
progession, either increasing or decreasing, is found by 
multi[)lying the last term by the ratio, and then dividing 
the diflerence of this product and the first term by the 
difference between the ratio and unity. 

Thus, in the series 2, 4, 8, 16, 32, 64, 128, 256, 312, 

we shall have ^^1^^~^ = 1024-2=1022, the sum of 
2—1 

the terms . 

Or the same rule, without considering the last term 
may be expressed thus : 

Find such a power of the ratio as is denoted by the 
number of terms of the series ; then divide the diflference 
between this power and unity, by the difference between 
the ratio and unity, and the result, multiplied by the fir9^ 
term, will be the sum of the series. 

Thus, in the series a-{-ar-fo»"^4-a»"'+a»'*,^c. tear"', 
we shall have 



AND PROGRESSION. 97 






Where it is to be observed, that if the ratio, or com- 
mon multiplier, r, in this last series, be a proper fraction, 
and consequently the series a decreasing one, we shall 
have, in that case, 

a 
a-{-ar-\-ar^ -\-ar^ ■\-ar'* , &c. nd infinitum = . 

9. Three quantities are said to be in harmonical pro- 
portion, when the first is to the third, as the difference 
between the first and second is to the difference between 
the second and third. 

Thus, a, 6, c, are harmonically proportional, when a : c 
y.a — b : b — c, or« : c\\b — a : c — h. 
And c is a third harmonical proportion to a and b, when 
ab 

^^ 2a-b' 

10. Four quantities are in harmonica! proportion, when 
the first is to the fourth, as the difference between the 
first and second is to the difference between the third and 
fourth. 

Thus, a, b, c, d are in harmonical proportion, when 
a : d : : a—b : c — d, or a : d : : b — a : d~c. And d is 
a fourth harmonical proportional to a, b, c, when d= 



ac 



-, in each of which cases it is obvious, that twice the 
2a— 6 

first term must be greater than the second, or otherwise 

the proportionality will not subsist. 

11. Any number of quantities a, b, c, d, e, kc. are in 
harmonical progression, if a : c : : a — b : 6 — c ; b : d : : 
b—c : c — d ; c : e : : c — d : d — e ; &c. 

, 12. The reciprocals of quantities in harmonical pro- 
gression, are in arithmetical progression. 

Thus, if a, b, c, d, e, &c. are in harmonical progrcs- 

1 1 1 1 1 „ M, , • ,- .• , 

sion, -, -, -, -J, -, &LC. will be m arithmetical progression. 



98 GEOMETRICAL PROPORTION, &c. 

13. An harmonical mean between any two quantities, 
is equal to twice their product divided by their sum. 

Thus, — r— = an harmonical mean between a and b*. 
a-j-b 

EXA^IPLES. 

1. The first term of a geometrical series is 1, the ratio 
2, and the numb«r of terms 10 ; what is the sum of the 
series. 

Here 1 X29 = l X512=512, the last term. 

. . 512X2-1 1024—1 ,_„ ^, . , 

And — - — - — = = 1023, the sum required. 

2 — 1 I ^ 

1 

2. 1 he first term of a eeometrical series is ^, the ra- 
j * 2 

tie --, and the number of terms 5 ; required the sum of 

the series. 

1 /'In" 1 1 1 
Here -X{ - ) =_x— =— -, the last term. 

2 \3/ 2 81 162 

1 1_VJ. ± 1_ 101 ^ 191 

And ^— i«=-ili=a— i^=_Xr=~, the sum. 
l-i I 243 2 162' 

3. Required the sum of 1,2, 4, 8, 16, 32, &c. conti- 
nued to 20 terms. Ans. 1048675. 

4. Required the sum of 1, -, -, -, — , — , &c. continu- 

^ ' 2' 4 8' 16' 32' J27 

ed to 8 terms. Ans. 1— - 

128 

5. Required the sum of 1, -, -, — , — , &:.c. continued 

3 9 27 81 9841 

to 10 terms. Ans. 1 ^,,^^. 

19683 

6. A person being asked to dispose of a fine horse, said 
he would sell him on condition of having a farthing for 

* In addition to what is here said, it may be obsen'ed, that the ratio •£ I .i 
two squares is frequently called duplicate ratio; of two square roots, s^ib- I " 
dvplicate ratio ; of two cubes, triplicate ratio ; and of two cube roots, sub- I '^ ^m 
triplicate ratio ; &c. I lijj. 



Of Equations. 99 

the first nail in his shoes, a half-penny for the second, a 
penny for the third, twopence for the fourth, and so on, 
doubling the price of every nail, to 32, the number of nails 
in his four shoes ; what would the horse be sold for at 
that rale ? Ans. 4473924/. 6s. 3^d. 



Of equations. 

The DocTRfNE of Equations is that branch of algebra, 
which treats of the methods of determining the values of 
unknown quantities by means of their relations to others 
which are known. 

This is done by making certain algebraic expressions 
equal to each other (which formula, in that case, is called 
an equation), and then working by the rules of the art, till 
the quantity sought is found equal to some given quantity 
and consequently becomes known. 

The terras of an equation are the quantities of which 
it is composed ; and the parts that stand on the right and 
left of the sign =, are called the two members, or sides, of 
the equation. 

Thus, if x=a-\-b, the terms are x, a, and b; and the 
meaning of the expression is, that some quantity x, stand- 
ing on the left hand side of the equation, is equal to the 
sum of the quantities a and b on the right hand side. 

A simple equation is that which contains only the first 
power of the unknown quantity : as, 

x-\-a=3b, or ax=.hc, or 2x-\-Za^ ^^bb^ ; 
Where x denotes the unknown quantity, and the other 
letters, or numbers, the known quantities. 

A compound equation is that which contains two or more 
different powers of the unknown quantity ; as, 
x^-{-ax—b, or a;3 — 4x2 4-3x=25. 

Equations are al?o divided into different orders, or re- 
ceive particular names, according to the highest power of 
the unknown quantity contained in any one of their terms : . 
as quadratic equations, cubic equations, biquadratic equa- 
. tions, &c. 



100 Of equations. 

Thus, a quadratic equation is that in which the unknown 
quantity is of two dimexisions, or which rises to the second 
{(ower : as, 

a-2=20; x2-{-cix=b, or 3x2 -\-10x=100. 

A cubic equation is that in which the unknown quantity 
IS of three dimensions, or which rises to the third power : 
as, 

:r3=27 ; Sx^ — 3.t=35 ; or x^—ax'^+bx—c. 

A bi']uadratic equation is that in which the unknown 
quantity is of four dimensions, or which rises to the fourth 
power: as, a;«=25; 5a:* — 4x=^6 ; or x* — ax'^-^-bx^ — ex 

And i?o on for equations of the 5th, 6th, and other high- 
er orders, which are all denominated according to the 
highest power of the unknoivn quantity contained in any 
one of iheir terms. 

The root of an equation is such a number, or quantity, 
us, being substituted for the unknown quantity, will make 
lioth sides of the equation vanish, or become equal to each 
other. 

A simple equation can have only one root ; but every 
compound equation has as many roots as it contains di- 
mensions, or as is denoted by the index of the highest 
power of the unknown quantity, in that equation. 

Thus, in the quadratic equation a;3_|_2x=15, the root, 
or value of a;, is either + -'^ or — 6; and, in the cubic 
equation x^ — 9x -j-26x=24, the roots are 2, 3, and 4, as 
Will be found by substituting each of these numbers for x. 

In an equation of an odd number of dimensions, one of 
its roots will always be real ; whereas in an equation of 
an even number of dimensions, all it roots ma}' be imagin- 
ary ; as roots of this kind always enter into an equation 
by pairs. 

Such are the^quations a;^ — 60;+ 14=0, and a;* — Sx^ - 
9x2 4yjx + 50=0*. 



• See, for a more particular account of the general theory of equations, 
further on, or Vol. II. of Bomiycastle's Treatise on Algebra, 8vo. 1820. 



SIMPLE EQUATIONS, &c. 101 

OF THE 

RESOLUTION ofSIMPLE EQUATIONS, 

Containing only one unknown Quantity. 

yhe. resolution of simple, as well as of other equations, 
is the disengaging the unknown quantity, in all such ex 
pressions, from the other quantities with which it is con 
nected, and making it stand alone, on one side of the equa- 
tion, so as to be equal to such as are known on the other 
side ; for the performing of which, several axioms and 
processes are required, the most useful and necessary of 
which are the following :* 

CASE I. 

Any quantity may be transposed from one side of an 
equation to the other, by changing its sign ; and the two 
members, or sides, will still be equal. 

Thus, if x-|-y=7 ; then will x=7 — 3, or x=4. 

And, if x--4 + 6 = 8 ; then will a;=84-4 — 6=6. 

Also, if x—a-{-b=:c—d: then will x=a — b-^c~ d. 

And, if 4a;-8=3a;+20; then 4a:- 3x=20+8, and con- 
sequently x=9.8. 

From this rule it also follows, that if a quantity be found 
on each side of an equation, with the same sign, it may 



* The operations required, for the purpose here mentioned, are chiefly such 
as are derived from the following- simple and evident principles : 

1. If tlie same quantity be added to, or subtracted from, each of two equal 
quantities, the results will still be equal ; which is the same, in effect, as tak- 
ing any quantity from one side of an equation, and placing it on the other 
side, with a contrary sign. 

2. If all the teniis of any two equal quantities, be multiplied or divided, . 
by the same quantity, the products, or quotients thence arising, will be equal. 

3. If two quantities, either simple or compound, be equal to each other, 
any like powers, or roots, of them will also be equal. 

All of which axioms will be found sufliciently illustrated, by the processes 
arising out of the several examples annexed to the si& different cas»s given 
in the text. 

K 2 



102 SIMPLE EQUATIONS. 

be left out of both of them ; and that the signs of all the 
terms of any equation may; be changed from + to — , or 
from — '^to 4", without aUering its value. 

Thus, if x-H5=7+6 ; the.n, by cancelling, x=7. 

AticJ,:if:cfr— ^=="6. r;c ; then, by changing the signs, a;— 
a=c — b, or x=^a-\-'c^b. ' ' ' 

EXAMPLES FOR PRACTICE. 

1. Given 2a;+3=a;4- 1 7 to find x. Ans. a:=14. 

2. Givea 5x — 9=4a:4-'7 to find a:. Ans. a;— 16. 

3. Given ar+9 — 2=4 to find x. Ans. a;=— 3. 

4. Given 9a: — 8=8x — 5 to find x. Ans. x=3. 

6. Given 7x+S— 3=6x-{-4 to find x. Ans. .x=— I. 

CASE II. 

If the unknown quantity, in any equation, be multiplied 
by any number, or quantity, the multiplier may be taken 
away, b}^ dividing all the rest of the terms by it ; and if it 
be divided by any number, the divisor may be taken away, 
by multiplying all the other terms by it. 

c 

Thus, if ax=2ab—c ; then will a;:=36 . 

a 

And, if 2x+4=16 ; then will x+2=0, 

orx=8-2=6. 

Also, if 1=5+3 ; then will x=10+6=16. 

2x 
And, if ——2=4 ; then 2x — 6=12, or, by division, 

,:-.3=6, ora;=9. 

EXAMPLES FOR PRACTICE. 

1. Given 16x-f-2=34 to find x. Ans. x=2. 

2. Given 4x — 8= — 3x+ 13 to find X. Ans. x=3., 

3. Given 1 Ox— 19=7x4-17 to find x. Ans. x=12. 

4. Given 8x-3+9=-7x+9+27 to find x. 

Ans. x=2. 

Ad 
6. Given ^ax - Zah—Ud. Ans. xt=b-\ — r- 

0.0 



SIMPLE EQUATIONS. 103 



CASE III. 



Any equation may be cleared of fractions, by multiply- 
ing each of its terms, successively, by the denominators of 
those fractions, or by multiplying both sides by the product 
of all the denominators, or by any quantity that is a mul- 
tiple of them. 

Thus, if--l — =5, then, multiplying by 3, we have x-{- 

3x 

— = 15 ; and this, multiplied by 4, gives 4a; 4" 3« = 60 ; 

4 

60 4 
'Whence, by addition, 7a:=60, or x=—=8- 

X X 

\ And, if — 1--=10 ; then, multiplying by 12, (which is a 
4 6 

multiple of 4 and 6,) 3a:-}-2x=120, or 5x=120, or x= 

=24. 

5 

It also appears, from this rule, that if the same number, 
or quantity, be found in each of the terms of an equation, 
either as a multiplier or divisor, it may be expunged from 
all of them, without altering the result. 

Thus, if ax=^ab-\-ac ; then by cancelling, x=b-]-c. 

X h c 
And, if — I — =- : then a;+6=c, or x=c — b, 
a a a 



EXAMPLES rOR PRACTICE. 

3a; X 

1 . Given —=—f 24 to find X. Ansx=19i. 

2 4 * 

2. Given J+^-f ^=62 to find x. Ans. x=60. 

3 5 2 

3. Given ——-f -=20 ^^ to find x. 



Ans. x=29f 



104 SIMPLE EQUATIONS. 

4. Given ^l+d:!=i6— ^- to find x. 



x+a X 2a; a+6 

5. Given — ; 1 — = 1 -y- to find x 

oca a 



Ans. x=13. 
a^ch — ach^ —a^cd 



acd-\-abd — 2cbd 

CASE IV. 

If the unknown quantity, in any equation, be in the 
form of a surd, transpose the terms so that this may stand 
alone, on one side of the equation, and the remaining 
terms on the other (by Case 1) ; then involve each of the 
sides to such a power as corresponds with the index of 
the surd, and the equation will be rendered free from any 
irrational expression. 

Thus, if .^x— 2=3; then will ^x=34-2=6, or, by 
squaring, x=52=25. 

And if v^(3a-+4) = 5 ; then will 3ar+4=25, or 3a;=25 

21 
— 4=21, or a- =—=7. 

Also, if V(2a^^+3) + 4=8 ; then will l/(2x-\-3)=Q~-4 
=4, or 2x+3=43=64, and consequently 2a=64 — 3=61, 

or x=-=30-. 



EXAMPLES FOR PRACTICE. 

1. Given 2^x4-3=9 to find x. Ans. a=9. 

2. Given ^(x+1)- 2=3 to find X. Ans. x=24. 

3. Given 3/(3x+4)4-3=6 to find x. Ans. x=7|. 

4. Given v/(4+x)=4— yx to find x. Ans. x^2i. 

5. Given ^(4a^-\-X') = X/(4b*+x*) to find x. 

(/,4_4a4\ 
~2~a2~ /• 



SIMPLE EQUATIONS. 105 

CASE V. 

If that side of the equation, which contains the unknown 
quantity, be a complete power, the equation may be re- 
duced to a lower dimension, by extracting the root of the 
said power on both sides of the equation. 

Thus, if x3=81 ; then a;=^81=9 ; and if x3=27, 
then a;=3/27=3. 

Also, if 30-2—9=24 ; then 3x2=24+9=33, or x^ = 

33 

— =11, and consequently a;=:^ll. 

And, if x2 4-6x+9=27 ; then, since the left hand side 
of the equation is a complete square, we shall have, by 
extracting the roots, a;+3= v^ 27=^/(9 X3)=3v/3, or a; 
=3^^3-3. 

EXAMPLES FOR PRACTICE. 

1. Given 9a;2— 6=30 to find a:. Ans. x=2. 

2. Given x-* +9=36 to find z. Ans., x=3. 

3. Given x^-\-x+}=— to find x. Ans. x=4. 

I o^ 1 ** 

4. Giyen x^-\-ax-\-——b'' to find x. Ans. x=& — x- 

4 ■^ 

5. Givenx2 + 14x+49=:_-121 to findx. Ans. x=4. 



CASE VI. 

Any analogy, or proportion, may be converted into an 
aquation, by making the product of the two extreme terms 
equal to that of the two means. 

Thus, if 3x . 16 : : 5 : 6 ; then 3xX 6=16X5, or 18x 

80 40 4 
^-80, or .=-=-=4-. 

2x 2cx 

And, if — : a : : 6 : c ; then will -^=ab,OT2cx=3ab; 
o o 



106 SIMPLE EQUATIONS. 

^. A- ■ ■ 3afc 

or, by division, a;= — . 

Also, if 12— X : ; : 4 : 1 ; then 12— x =— =2x, or 2x 
2 2 

4-a;=12 ; and consequently x= -^^^'^- 



EXAMPLES FOR PRACTICE. 

^.3 , , ^ , , 20ai 

1. Given -x : o : : 5oc : ca to nnd x. Ads. x= — 5- . 

4 3a 

2 

2. Given 10 — x : - x : : 3 : 1 to find x. Ans. x=S\. 
3 * 



3. Given 8+8x : 4x : : 8 : 2 to find x. Ans. x = l, 

4. Given x : 6— x : •. 2 : 4 to find a;. Ans. x=2, 

5. Given 4x : a : : 9^x : 9 to find x. Ans. x=-7:. 

16 



MISCELLANEOUS EXAMPLES. 

1. Given 5x— 15=2x+6, to find the value of x. 

Here 6x— 2x=6-f 15, or 3x=6+ 15=21 ; and there- 

21 
fore x= -x-=7. 

2. Given 40-6x- 16 = 120- 14x, to find the value of x 
Here I4x— 6x=120 -40+16 ; or 8x=136 — 40=96 ; 

and therefore x=— -=:12. 
8 

3. Given Sx^ — 10.t=8x4--'^^9 to find the value of x. 
Here 3x — 10=8-)-x, by dividing by x ; or 3x — x=:8-f' 

10=18, by transposition. 

18 „ 
And consequently 2x=18, or x=— =9. 

4. Given 6ax^ - 12abx^=-3ax^-\-6ax^ , to find the value 
of X. 



9' 



SIMPLE EQUATIONS. 107 

Here 2a; — 4b=x-{'2, by dividing by 3ax^ ; or 2a;— a:= 
2+46 ; and therefore x=4h-{-2. 

5. Given a;3 + 2x+ 1 = 16, to find the value of ». 
Here a;+l=4, by extracting the square root of each 

tide. 

And therefore, by transpositisn, .r=4 — 1=3. 

6. Given 5ax—3b=^clx-{-c, to tind the value of x. 

Here 5ax^2dx—c-\-3b ; or {5a— 2 d)x=c-}- 3b ; and 

c + 36 
therefore, by division. x= --. 

3^ X ST 

7. Given -A — = 10, to find the value of x. 

2 3 4 

Here x-— ^ =20 : and 3.r-2xH =60 : or Ux 

3 4 ' 4 ' 

— 8x+6x=240 ; whence 10x=240, or x=24. 
^.^3 -y X 19 

8. Given — — — [--=20 — , to find the value of x. 

2x 
■ Here x-3-1 =40-x+19 ;or3x— 9-|-2x=120 — 3x 

+57; whence 3x+2x+3x= 120+57+9 ; that is 8x= 
186, or x=23i. 

2x 

9. Given ^— — [-5=7, to find the value of x. 

2x 2x 

Here ^—=7 — 5=2 ; whence, by squaring,— =2^ = 

O %j 

4, and 2x=12, or x=6. 

2a2 

10. Given ^ + w'('a^+a;0 = . ; ^.^ to find the 

palue of X. 

Here x^{a^+x-)-\-a^-\-x~=2a~ ; orx^(a2+x ) == 

a2— x2, and x2(o2+x2)=:a4 -2a2x2+x* ; whence a^js 

+x*=^a'*— 2a3a-2+j;4, anda2xa=a4— 2a2x2 ; therefore 3 

a* a2 , , a2 

%^x^~a*, or X = — — ;^=-^; and consequently x—a/— 

J 3 a „ , . -. ■ 

=a V o^^'a/q "^■v/^* ti^e answer required. 

O I/O 



108 SIMPLE EQUATIOKS. 



EXAMPLES FOR PRACTICE. 



1. Given 3x — 2-1-24=31, to find the value of x. 

Ans. a;=3 

2. Given 4 — 9?/= 14 — 11?/, to find the value of y. 

Ans. 2/=5 

3. Given a;+18=3x— 5, to find the value of x. 

Ans. a;=ll 

4. Given x-\-'--\-~=ll, to determine the value of x. 

Ans. a:=( 

5. Given 2a:— -4-l=5a-— 2, to find the value of x. 

Ans. x= 

XXX 7 

6. Given -+- — = — , to determine the value of x. 



7. Given — - — | — =4—^ , to find the value of x 

2 ' 3 4 



Ans. x=l 



Ans. x=3 

8. Given 2-{-y/3x= v^4-|-5x, to find the value of x. 

Ans. x=l 

9. Given x-!-a= — ; — , to find the value of x. 

a-f-x 

Ans. x= — 
2(j 

10. Given v^x-}-»/a-|-x= — ; — ; — r to find the valu 

^{a+x) 

of X. Ans. x= 

„. ax~b , a bx bx — a ^ , , 

11. Given — , — +-=- , to find the valu 

^323 3^ 

of X. * Ans. X 



3a- S 



12. Given ^a^-{'X-=^\/b'*-}-x'^ , to find the value of : 

b*- 

Ans. x=^~— 

^ 2 a 



SIMPLE EQUATIONS. 109 



13. Given y/a-\-x-\-^a''X—^ax, to - find the value 
of X, Ans. a:=^j;^ 



a a 



14. Given — -, — I =6, to determine the value of x. 

l+a; 1 — X 

Ans, x=<y — 7 — 



15. Given a+x= ^ a2 +x V (6^ ■^x^)^o find the value 
of X. ^ Ans. a;=4^-« 

16. Given |y(xa+3a8)-iy(a;2-3a2)=a;v/a, to find 
the value of x. Ans. ^—Vj^^J^ 

17. Given v^(a+x)4-^(o-x)=&, to find the value of 
X. Ans a;=--v/(4a-6a) 

18. Giveni/(a+x) + VC<^— x)=6, to fi nd the value of ar . 

Ans.x=^a2-(^-3^; 

19. Given v/a+v/a:=^ax, to find the value of x. 

Ans. x=- 



(x-4~l\ /'x— 1\ 

- — T )+V'( "Xt) ~"' *^ determine the 

a 

value of X. Ans. x= — m==: 

v^a2_4 

21. Given v'(a2-f-fla^)=a—^(a2 —ax), to find the value 

of X. k ^ jc, 

Ans. x=-^3. 

22. Given ^(a2-.x2)+iV("- " 0=«''a/(1 -*0. *<> 

find the value of x. » _* . / «'^ - 1 \ 

Ans. X- ^^^—^^ 



* Bonnycastle, in his Key, sajs this answer is wrong; the answer which 

he gives, is / — . ; Now these two answrrs give the same value fOi 

♦ ' L 



no SIMPLE EQUATIONS. 

23. Given ^(^x-i-a)=c—y/{^x-{-b), to find the value of .r. 

h c 4fec 

24. Given -/ — p- +y = y , to find the 



value of X. Ans. a; 



Of the resolution of simple equations., containing two 
unknown quantities. 

When there are two unknown quantities, and two inde- 
pendent simple equations involving them, they may be re- 
duced to one, by any of the three following rules ; 



RULE I. 

Observe which of the unknown quantities is the least 
involved, aud find its value in each of the equations, by 
the methods already explained ; then let the two values, 
thuo found, be put equal to each other, and there will arise 



x; that is, x=y (-^1-^), or^ ^^ ^ ^ 2a2— 3) ' because if we assume any 

value for a, and substitute it in both, ^ve get the same value for a;. Let cc=2, 
, ,a2 — K 3 , as — 1 3 ^9 

consequently both give the same answer. Or thus, b}' hypothesis ; Let 
y-rtS— l^ o2 — 1 

'v/v«2 4.3/'~'\/(a4 4.2a2— 3)' 
_ . a2 — 1 a4 — 2a2 4-l ^. . ,. 1 

By squannar -^ —- :, Dividins' by aa — 1 we eet = 

•' ^ ^02+3 a4-f.2o2— 3' ^ ^ ^ a2 -f-3 

a2 — 1 

— r —, and by multiplication c4 + 2a2 — 3=a4 4>2a2— 3. There- 

«4 -\-2a2 — 3 ' 

fore dif hypothesis is correct. 



SIMPLE EQUATIONS. Ill 

a new equation with only one unknown quantity in it, the 
value of which may be found as before.* 

EXAMPLES. 

1. Given \ „ y ,„ > to find the values of x and y. 
I 5x—2y=10 y ^ 

23 — 3« 
Here, from the first equation, x= — - — -^ 

. ^ r , , 10 + 2y 

And irom the second, a-= 



Whence we have 



5 

23—32/ 10+22/ 



2 5 ' 

Or 115-15i/=20+4t/, or 1%=115- 20=95, 

rru.- 95 ^ , 23-15 ^ 

That IS, «=-— =5, and x= =4. 

'^19 2 

2. Given ? ^*y^^ ( to find the values of x and y. 

Here, from the first equation, x=a—y^ 
And from the second, x=6+j/, 
Whence a — y=.h-\-y, or 2y=a—b^ 

And therefore y=———, and x=o — y, 
Or, by substitution, x=^a — ■ = —5—. 

3. Given <?T? o?to find the values of x and v. 

2t/ 
Here, from the first equation, a:=14 — ^, 

And from the second, x=24 -, 

2 

2« 3t/ 

Therefore, by equality, 14 — -|-=24 — ^, 



* This rule depends upon the well known axiom, that things which are 
equal to the same thing, are equal to each other ; and the two follovving 
methods are founded on principles which are equally simple and obvious. 



112 SIMPLE Equations. 

And consequently 42— 2t/=72 -^ 

Or by multiplication 84 — 4i/=144 — %; 

And, therefore, also 52/= 144 — 84=60, 

60 24 

Or, by division, a;=— =12, andx=14 — 5-=6' 

EXAMPLES FOR PRACTICE, 

1. Given 4a-+j/=34, and 4t/ + a:=16, to find the values 
of X and y. ■ Ans. x=8, j/=2. 

2. Given 2x-{-3y=16, and 3x-2y=ll, to find the va- 
lues of X and y. Ans. x=5, 2/=2. 

3. G.ven -+f =-, and _+-|=_ to find the va- 
lues of X and ?/. Ans. a=^, y=i. 

4. Given | f^j^g^"^ 5 to find x and y. 

Ans. x=a4"*, and y=^a — Jft. 

(-+1=0 

1 2 3 r 

5. Given < > to find x and y. 

V.3 2 J Ans. x=12, andy=6. 



w -+^=9 
6. Given < 2 3 ^ to find x and y 



x:?/::4:S) Ans. x=12,andy=9. 

2x 3i/ 
T. Given x+2/==80, and -^=-j^, to find x and ?/. 

N Ans. x=42y\, and j/^STfj. 

8. Given ^ — 6=-, and x=::y-\-&, to find x and y, 

Ans. xc=24, and y=\Z. 



RULE II. 



Find the value of either of the unknown quantities in 
that equation in which it is the least involved ; then sub- 
stitute this value in the place of its equal in the other 



SIMPLE EQUATIONS. 113 

equation, and there will arise a new equation with only one 
unknown quantity in it ; the value of which may be found 
as before. 

EXAMPLES. 

1. Given \ /"^^^^l^ I to find the values of x and y. 

From the first equation, x:^l 7— 21/ ; which value, being 

substituted for x, in the second, 

gives 3{\7—2y) — y=2. 

Or 5l--6y-y=2, or 7i/=3 1 — 2=49, 

49 
Whence y =—=7, and x=\'7^2y=3. 

2. Given 5 ^+2/=^^ ( to find the values of x and y. 

I X — y= -i i 

From the first equation, x=\3—y ; which value being 
substituted for a;, in the second, 

Gives l3-y-y=3, or 2y=\3—5=\0, 

Whence ?/= — =6, and a:=13 — y=^- 

3. Given ^ ^ ' i' , '■,"'* Mo find the values of x and y. 

\ x^ -f-y^ =c ) 

Here the analogy in the first, turned into an equation, 

gives bx=ay, or x= — 
And this value, substituted for x in the second, 

giyes(^y+y'^=c,or^^-{-y^=c, 

b3c 



Whence we have a'^ y^-\-b''y^=zb^c, or y''^-^^^ 

c c 

And, consequently, y^^V-Tj^^ ^°^ ^= ^^-^TI^-^ 



EXAMPLES FOR PRACTICE. 



1. Given -+72/=99, and|4-7ic=51, to find the values 

♦f X and y. Ans, x=7, and Sf=14. 

. L.2 



114 SIMPLE EQUATIONS. 

2. Given ^- 12-^+8, and -4^+1-8=^ -+27, 

-6 4 bo 4 

to find the values of x and y. Ans. x^60, i/=40. 

3. Given a;+j/=s, and x^-j-yS—.^;^ to fju^j tjjg values of 

. ■ s'-\-d s2— d 
jc and y. Ans. a;= , y= . 

4. Given Sx— 3j/=150, and 10a:+152/=:825, to find* 
and y. Ans x==45, and y=25>. 

5. Given x-{-y=\6, and x : y : : 3 : 1, to find a; and y. 

Ans. x=12, and 2/=4. 

6. Given x+ |=12, and 7/-l--=9, to find x and y. 

Ans. .'c=10, and y=z4. 

7. Given x : y : : 3 : 2, and x2 - i/2 =20, to fiod x and y. 

Ans. x=6, and y=4. 

8. Given|-12=|l3+,and^+|+16=?^ + 27, 
to find X and y. Ans. x=60, and 2/=20. 

RULE in. 

Let one or both of the given equations be multiplied, or 
divided, by such numbers, or quantities, as will make the 
term that contains one of the unknown quantities the same 
in each of them ; then, by adding, or subtracting, the two 
equations thus obtained, as the case may require, there will 
arise a new equation, with only one unknown quantity in it, 
which mav be resolved as before*. 



* The values of the unknown quantities in the two literal ax -^/)3/=c, and 

a'a;-|-6'y=c', may be found in general teims, by multiplying- tlie first bj' a', 

and the second by a, and then working- according to die last rule, when the 

1 T • 1 -.1 , <»c' — CO' - cb' — bd , . , , . 

results, so determined, will be y= ~ r— -, and x=—, pr; which solution 

ab —ba ab — ba 

may be applied to any particular case of this kind, by substituting the nume- 
ral of a, h, a', b', in the place of the letters, and observing, when either of 
them is negative, to change the signs accordingly. 

"WTiere the numerator is the difference of the pi-oducts of the opposite co- 
efficients in the order in which y is not found, and tlie denominator is the dif- 
ference of the products of the opposite coefficients taken from the orders that 
involve the two unknown quantities. Coefficients are of the same order which 



SIMPLE EQUATIONS. 115 



EXAMPLES. 

1. Given j " Tg ^i . ? to find the values of x and y. 

First, multiply the second equation by 3, and it will give 
3x + 6y=42. 

Then, subtract the first equation from this, and it will give 
%~5i/=42— 4G, ori/=2. 

Whence, also, a;=14 — 2j/==14— 4=10. 

< 2. Given \ „ X^s — ifi i ^° ^'^^ ^^^ values of a; and y. 

Multiply the first equation by 2, and the second by 5 ; 

then lOx— 6?/=18, and igx-f-26?/=80. 

And if the former of these be subtracted from the latter 

, 62 

there will arise 31?/=62, or y= — =2. 

O 1 

Whence, by the first equation, x=: -= — =3. 

5 6 

EXAMPLES FOR PRACTICE. 

1. Given ^-- — 1-6^=21, and ^i-=23-6x, to find a; 
and y. Ans. x=4, and y=3. 

I 2. Given 3x+72/=79, and 22/=9+-, to find x and ?/. 

Ans. x=10, and y=l. 

3. Given 30x+40?/=270, and 50a-f-30i/=340, to find 
K and y. Ans, a;=&, and y=3. 

4. Given Sx — 3i/=2a;+2i/, and x+y : xi/ : : 3 : 5, to 
find X and y. Ans. x=10, and i/=2. 

5. Given x2i/+ay=30, and x^-{-y^=^Sb, to find x 
and y. Ans. x=3, and y=2. 



jiAer aiTect no unknown quantity, as c and c': or the same unloiown quantity 
n the ditlerent equations, as a and a'. Coefficients are opposite when they 
iffect tlie different unknown quantities in the different equations, as a and b', 
>('«nd b. 



116 SIMPLE EQUATIONS. 

6. Given ^=~? ^' ^"^ ^ 4^2'^3' 

find X and y. Ans. a;=12, and y=6, 

7. Given x+J/ : a : : x-y : b, and x^ —y^=c, to fine 
the values of a; and y. 

a+6' , c a — 6 

Ans. «:=-^v/^, '-'^-T-^aj 

8. Given ax-{-by^c, and dx+ey=f, to find the value: 

of ic and y. 

ce — bf af-di 

Ans. a: = —, y= r 

ae-~bd ac-~b( 

9- Given x-\-y=a, and a;^ —y^=.b, to find the values o 



2a '' 2a 

10. Given a;- -j-^2/=«5 and i/2 +arj/=6, to find the valuei 

of X and y. k ^ ^ 

^ Ans. a:= — - — r-riV'^—rr — TT 

Of the resolution of simple equations, containing three o: 
more unknown quantities 

When there are three unknown quantities, and threi 
independent simple equations containing them, they ma; 
be reduced to one, by the following method*. 

RULE. 

Find the values of one of the unknown quantities i 
each of the three given equations, as if all the rest wer 



* The necessity for observing that the given equations, in this and othf 
similar cases are so proposed as to be independent of each other, wfil b 
obvious from the tbllowing example : 

a;-2y+2=5;2a+y_«=7; x + 3y— 2z=2; 
where, if it were required to determine the values of x, y and z, it will l 
found by eliminating a from each of them, and then equating the results, th-l 

5y— 3z=— 3, and 5y— 32^— 3; 
which equations, being identical, or both the same, furnish no determina 
answer. And, in efl'ect, if the three equations be properly examined, it wi 
be found, that the third is merely the difference of the first and second, an «« 
consequently involves no condition but what is contained in the other two. 



m 
M 
mi 



»T 
ml 



s 

tet 



SIMPLE EQUATIONS. 117 

cnown ; then put the first of these values equal to the se- 
;ond, and either the first or second equal to the third, and 
here will arise two new equations with only two unlcnown 
juantities in them, the values of which may be found as in 
he former case ; and thence the value of the third. 

Or, multiply each of the equations by such numbers, or 
juantities, as will make one of their terms the same in 
hem all ; then, having subtracted any two of these result- 
ng equations from the third, or added them together, as 
he case may require, there will remain only two equa- 
ions, which may be resolved by the former rules. 

And in nearly the same way may four, five, &c. un- 
|:nown quantities be exterminated from the same nutn- 
>er of independent simple equations ; but, in cases of this 
and, there are frequently shorter and more commodious 
oethods of operation, which can only be learnt from prac- 
ice*. 



EXAMPLES. 



1. Given { x+2y-\-3z^62 \ to find x, y, and 



» The values of the imknown quantities in the three literaj equations 
ax + iy + cz=d; a'x + b'y+ c'2=d'; a"x ■^b"y + c"z=d" ; 
nay be exhibited, in general terms, like those before mentioned, as follows ; 
__ db'c" —dcb'' + cdb'—bdc' + hc'd'—ch'd" 
^~ ab'c"—acb''+ca'b'—ba'c''.-^bc'a''—cb'a" 
ad'c' ' — at d"-\- c a'd" — da'c"-i-dc'a" — cd'a'' 
*'^a6'c — ac'b" -fca'b' — ba'c -j-bc'a' — cb'a'' 
ab 'd'' — a'b"-^ dab ' — b a^d" 4- bd'a — db'a" 
^ ab c" — ac' b' ' -f ca'b'' — ba'c" -f- bed ' — cb'a ' 
vhich formula, by substitution, may be employed for the resolution of any 
lumeral case of this kind, as in the instance of two equations before given. 
The numerator of any of these equations, such as 2, consists of all the dif- 
erent products, which can be made of three opposite coeflRcients taken frem 
he orders in which 2 is not found; and the denominator consists of all the 
products that can be made of the tliree opposite coefficieuts taken fcoa tint 
orders which involve the three unknown quantities. 



118 SIMPLE EQUATIONS. 

Here, from the first equation, x=29 — y-2. 

From the second, x=62 — 2y~ Sz, 

n i 
And from the third, x—20—-u z 

Whence 29— ?/ — 2r=:62-2?/-32-, 

2 1 
And, also, 29 -y^z =20— -y z. 

From the first of which y=33-2z. 

And from the second, 2/=27 z, 

2 
3 
Therefore 33— 2z=27 — z, or 2=12, 

Whence, also, 2/=33 — 22'=9 
And a; =29 — y-.z=^. 
i2x■\■^y-^2,z=^22\ 
2. Given } 4x^2y+5z= 18 ) to find re, ;y, and z. 

(Sx+ly-z ^63) 
Here, multiplying the first equation by 6, the second by 
3, and the third by 2, we shall have 

12a;4-247/-18z=132, 
12.T- Cy+\5z=54, 
I2x+\4y~ 2^=126. 

And, subtracting the second of these equations succes- 
sively from the first and third, there will arise 
• 30y^332=7Q, 
20?/— 17z=-72. 

Or, by dividing the first of these two equations by 3, 
and then multiplying the result by 2, 

20y — 22z=52, 
20y—nz=12. 

Whence, by subtracting the former of these from the 
latter, we have oz=^ 20, or z= 4. 

And, consequently, by substitution and reduction, 
y= 7, and x= 3. 

. 3. Given x+y-\-z= 53, x+2y-{-3z=l05, and .r+3jf+ 
Az= t-24, to find the values of x, y, and z. 

Ans. a-=24, y=6, and2=23. 



SIMPLE EQUATIONS. ](§ 

« 



y-{--z=l2, to find the values of x, y, and z. 

Ans. x=12, 2/= 20, ^=30. 

5. Giren lx+5y+2z=19, 8x+7y-{-9z = 122, and x+ 
4y+52'=56 to find the values of x, y, and z. 

Ans. x—4,y=9,z=3. 

6. Given a;+i/= a, x-{-z =h, and y+^ =c, to find the va- 
lues of X, y, and z. 

Ans. ,=^+£,.="+1=5 and .=^±i. 

..Given 1+1+1=6., ^+| + i=47.a„a|+|+l. 

J8, to find X, y, and z. Ans. x=24, 2/=60, and z = l20. 

8. Given z+2/=a;+100, i/-2x=2z - 100, and .?-f-lOO 
=3x-\-3y, to find x, y, and z. 

! ^ Ans. a:=9JT, 2/=45-j-V, and ^=63/-!-. 

9. Given x-\-y-\-z= 7, 2j;— 3 =^y+3z, and 5x + 5^= 3y-{- 
19, to find a;, y, and e. Ans. a;=:4, y=2, and 2'=]. 

^- 10. Given3a- + 5!/-4z=2e, 5a;— 2j/4-3z=46, and37/-|- 
ii'— a;— 62, to find x, y, and z. 

Ans. a;=J, ?/=:8, and z=9. 
11. * Givena:+2/4-2'=13,a;-f?/+«=17, a:+^+ii=18, 
ind t/-f2-f-M=21, to find .r, y, z, and u. 

Ans. x=2, y=5, 2-= 6, and «= 10. 

MISCELLANEOUS QUESTIONS, 

PRODUCING SIMPLE EQ,UATI0NS. 

The usual method of resolving algebraical questions, 
8 first to denote the quantities, that are to be found, by x, 
; or some of the other final letters of the alphabet ; 



* This can be resolved by proceacling- after the same manner, as equations 
ivolvmg three nnknown quantities : but the rcsotufion of it may be greatly 
icilitated, by introducing into the calculation, beside the principal unknown 
uantities, a new unknown quantity arbitrarily assumed, such as, for example, 
le sum of all the rest : 4||d when a little practised in such calculation';, they 
ecome tasv. ■' 



ISO SIMPLE EQUATIONS. 

then, having properly examined the state of the qaes 
lion, perform with these letters, and the known quanti 
ties, by means of the common signs, the same opera 
tions and reasonings, that it would be necessary to mak 
if the quantities were known, and it was required to veri 
fy them, and the conclusion will give the result sought. 

Or, it is generally best, when it can be done, to denot 
only one of the unknown quantities by x or y, and the 
to determine the expression for the others, from the na 
tare ©f the question ; after which the same method ( 
reasoaing may be followed, as above. And, in some case: 
the substituting for the sums and differences of quantities 
or availing ourselves of any other mode, that a prope 
consideration of the question may suggest, will greatly fj^ 
cilitate the solution. 

1. What number is that whos€ third part exceeds ii 
fourth part by 16? 

Let x= the number required. 

Then its - part will be - x, and its - part - x. 
3 ^ 4 4" 

And therefore - a;— -a;=16, by the question, 

3 

That is X x=48, or 4x— 3a:=192, 

4 
Hence x=l 92, the number required. 

2. It is required to find two numbers such, that thej 
sum shall be 40, and their difference 16. > 

Let X denote the least of the two numbers required, 

Then will x4-16 = to the greater number, 

And x+x-f 16=40, by the question, 

24 
That is 2x=40- 16, or x= -^=12= least number 

And x+ 1 6=1 2+ 1 6=28=the greater number require< 

3. Divide 1000/. between a, b, and c, so that a sha 
have 111. more than b, and c lOOZ. more than a. 

Let x=b's share of the given sum, 
Then will x+72=a's shar^ 
And x+172=c's share. 



1^ 



SIMPLE EQUATIONS. 121 

Henoe their sum is a;4-a-+72+x4-172. 

Or 3a;4-244=]000, by the question, 

That is 3x=1000- 244=756, 

756 
Or x= =252Z. =b's share, 

Hence x+72==324L=a.'s share. 
And x+172=424L=c.'s share. 
Also, as above, 2621. =b.'s share. 



Sum of all =1000Z. the proOf. 

4. Tt is required to divide lOOOL between two persons, 
so that their shares of it shall be in the proportion of 7 to 9'. 

Let 0,-= the first person's share, 
Then will 1000-ar= second person's share. 
And X : 1000— x : : 7 : 9, by the question. 
That is 9a=( 1 000 - x) X 7=7000 - 7x, 

7000 «„, ,^ , ^ , 

Or 9x-f 7x=7000, or x = —-5- = 437/. 10s. = 1st share, 

lb 

and 1000- x=1000- 437/. 10s. = 562Z. 10s. = 2d share. 

5. The paving of a square court with stones, at 2s. a 
yard, will cost as much as the enclosing it with pallisades, 
at 5s. a yard ; required the side of the square. 

Let x= length of the side of the square sought, 
Then 4x= number of yards of enclosure, 
And x^= number of yards of pavement. 
Hence 4xX5=20x= price of enclosing it, 
And x2 X 2=2x2=: the price of the paving, 
Therefore 2r2=20x, by the question, 

Or 2x=20, andx=10, the length of the side required. 

6. Out of a cask of wine, which had leaked away a 
third part, 21 gallons were afterwards drawn, and the cask 
being then guaged, appeared to be half full ; how much 
did it hold ? 

Let X = the number of gallons the cask is S!jpposed to 
have held. 

Then it would have leaked away -x gallons. 

•5 
M 



122 SIMPLE E(^UAT10NS. 

Whence there had been taken out of it, altogether, 
21+-X gallons, 

And therefore 2\-\--x=~x, by the question, 

3 
That is 63+x =-x, or 126-f-2a;=3.r, 

Consequently 3x — 2x 126, or a;=l26, the number of 
gallons reqaired. 

7. What fraction is that, to the numerator of which if 1 

be added, its value will be -, but if 1 be added to the do- 

nominator, its value will be -. 

4 

X 

Let the fraction required be represented by-, 

Then =-, and ', . =-, by the question. 

y 3' 2/+1 4' -^ ^ 

• Hence 3a;4-3=t/, and 4x=y-\- 1 , or x= , 

Therefore 3('^~^+3=y, or 32/+34-12=4y, 

r^, ■ ,. , y+I IS-fl 16 , 

That IS y=lo, and «= , ■= — - — =—-=4, 
^ 4 4 4 

4 
Whence the fraction that was to be found, is — . 

?5 

8. A market woman bought in a certain number of eggs 
at 2 a penny, and as many others at -3 a penny, and having 
sold them out again, altogether, at the rate ©f 6 for 2d.. 
found she had lost 4c?. ; how many eggs had she ? 

Let a;=the number of eggs of each sort, 

Then will-x=the price of the first sort. 

And ^3r= the price of the second sort. 

But 5 : 2 : : 2x (the whole number of eggs) : -r, 

o 



SLMPLE EQ.UATIONS. 123 

4 ,- 
\Vhence-T-=--the price of both sorts, when mixetl toge- 

b 
iher at the rate of 5 for 2d., 

And consequenly -x+-a; =4, by the question, 

2 3 o 

; That is 15a;+10x — 24x= 120, or a;— ISO, the number 
of eggs of each sort, a? required. 

9. If A ^^n perform a piece of work in 10 days, and b 

I in 13 ; in what time will they tinish it, if they are both set 
about it together ? 

Let the time sought be denoted by x, 

Then —= the part done by a in one day, 
And ■— = the part done by b in one day, 

X CO 

Consequently — -| — -=1 (the \vh®le work). 
That is 13x-f-10x=130, or 23a;=130, 
Whence a;= ——=5— days , the tiijc required. 

10. If one agent a, alone, can produce an effect c, in the 
time a, and another agent b, alone in the time b ; in what 
time will both of them together produce the same effect ? 

Let the time sought be denoted by x. 

Then a : e : : x : — = part of the effect produced by a. 
And 6 : e :: x : -r-= part of the effect produced by b. 

9 

€X ^X 

Hence — }— r=c (t^® whole effect) by the question, 
a 

X X 

Or — l"r=^ ^y dividing each side by e. 
Therefore x -\ — r-=a, or 6x+ax=a6, 



_ , ab . . , 

Consequently x== ——I = time required. 

U. How much rye at 4s. Qd, a bushel, must be mixed 



124 S5IMPLE EQUATIONS. 

with 50 bushels of wheat, at 6s. a bushel so that the "mix- 
ture may be worth 5s. a bushel ? 

Let x^the number of bushels required, 
Then 9x is the price of the rye in sixpences, 
And 600 the price of the wheat in ditto. 
Also (60+a;)X 10 the price of the mixture in ditto, 
Wheuce 9x4-600=500+ lO.r, by the question, 
Or. by transposition, lOx — 9a=600 — 500, 
Consequently a;=100 the number of bushels required. 

12. A labourer engaged to serve for 40 days, on con- 
dition that for every day he worked he should receive 
2Qd., but for every day he was absent he should forfeit 
^d., : now at the end of the time, he had to receive 
1/. lis. Sd. ; hosv many days did he work, and how many 
was he idle ? 

Let the number of days that he worked be denoted by 

X, 

Then will 40 — a; be the number of days he was idle, 
Also Wx the sum earned, and (40 — x) x 8. 
Or 320— 8a; the sum forfeited, 
Whence 20x-(320-8x) — 3Q0d.(=ll. lis. 5d.), by 

the question. 

That is 20x- 320+8.r=-380, 

Or 28a;=380+320 =700, 

Consequently x =—-=25, the number of days he 

worked, and 40— x=40~ 25=15, the number of days he 
was idle. 

q,DESTI0NS FOR PRACTICE. 

1. It is required to divide a line, of 15 inches in length, 
ioto two such parts, that one may be three fourths of the 
other. -Alls. 8a and 6^ 

2. My purse and money together are worth 20s. and 
the money is worth 7 times as much as the purse, how 
much is there in it ? Ans. l^s. 6d. 

3 A shepherd, being asked how many sheep he had in 



SIMPLE EQUATIONS. 125 

his flock, said, if I had as many more, half as many more, 
and 7 sheep and a half, I sho ild have just 600 ; how many 
had he ? Ans. 197. 

4. A post is one fourth of its length in the mud, one 
third in the water, and 10 feet above the water, what is- 
its whole leni^th.? • Ans. 24 feet 

5. After paying away - of my money, and then - of the 

4 5 

remainder, I had 72 guineas left ; what had I at tirst?' 

Ans. 120 guineas 

6 It is required to divide 300/ between, a. b, imd c, 

so thrit A may have twice as much as b, and c as mwh as 

A aad B together. Ans. a l(to/.. b 50/., c ibOl. 

7. A person, at the ti-rie he w<is married was ^^ times 
as old as his wife ; but after they had lived together 15 
years, he '.va^ only twice as old ; wh.it were their ages on 
their wedding day ^ 

, "". Ans. Bride's age lf>, bridegroom's 45 

8. What number i^ that from which, if 5 be subtracted, 
two thirds of the remainder will be 40 ? Ans 65 

9. At a certain election. 12P6 persons voted, and the 
succe-pful candidate had a majority of 120 ; how many 
voted for each ? 

Ans. 70H for one, and 588 for the other 

!0. A's age is double of b's, and b's is triple of c's, 

and the sum of all their ages is 140; what is the age of 

each ? Ans. a's 84, b's 42. and c's 14 

11. Two persons a and b, layout equal sums of mo- 
jiey in trade ; a gains li't;/. and b loses 87/., and a's money 
is now double of b's what did e'ach lay out ? 

Ans. 300/. 

12. A person bought a chaise, horse and harness, for 
60?. ; the horse came to twice the price of the harness, 
and the chaise to twice' the jfrice of the horse and har- 
nes.s : what did he give Tor each ? 

Ans. 13/. 6s. 8d.for the horse, 6/. 13s. 4d, 
for the harness, and 40/. for the chaise 

13. A person was desirous of giving 3d. apietee to some 
beggars, but found he had not money enough in his pocket 

M 2 



12€ SIMPLE EQUATIONS. 

by 8c?., he therefore gave them each 2c?., and had then Scfe 
remaining ; required the number of beggars ? 

Ans. 11 

14. A servant agreed to Hve with his master for 0»?. a 
year, and a livery, but was turned away at the end of 
seven months and received only ^l. I3s. Ad. and his livery ; 
what was its value ? Ans. 4/. 16s. 

15. A person left 560Z. between his son and daughter, 
in such a manner, that for every half crown, the son 
should have, the daughter was to have a shilling; what 
were their respective shares ? 

Ans. Son 400/., daughter 160/. 

16. There is a certain number, consisting of two places 
of figures which is equal to four times the sum of its di- 
gits ; and if 18 be added to it the digits will be inverted ; 
what is the number ? Ans. 24 

17. Two persons, a and b, have both the same income ; 
A saves a fifth of his yearly, but b, byspepdii.g 50/. per 
annum more than a, at the end of four years, finds himself 
100/. in debt ; what was their income ? 

Ans. 125/. 

18. When a company at a tavern came to pay their 
reckoning, they found, that if there had been three persons 
more, they would have had a shilling apiece less to pay, 
and if there had been two less, they would have had a shil- 
ling a piece more to pay ; required the number of persons, 
and the quota of each ? 

Ans. 12 persons, quota of each 5s. 

19. A person at a tavern borrowed as much money as 
he had about him, and out of the whole spent Is. ; he 
then went to a second tavern, where he also borrowed as 
much as he had now about him, and out of the whole 
Spent Is. ; and going on, in this manner, to a third and 
fourth tavern, he found, after spending his shilling at the 
latter, that he had nothing left ; how much money had he 
*t first? Ans. lUc/. 

20. It is required to divide the number 75 into two such 
parts, that three times the greater shall exceed seven 
times the less by 15. Aas. 54 and 21 



SIMPLE EqtJATlONS. 127 

2t. In a mixture of British spirits and water, i of the 
whole plus 2d gallons was spirits, and ^ part minus b gal- 
lons was water ; how many gallons were there of each ? 

Ans. Hb of wine, and 35 of water 

22. A bill of 120/. wa's paid in guineas and moidores, 
and the number of pieces of both sorts that were used 
were just 100 ; how many were there of each, reckoning 
the guinea at 21s., and the moidore at 27s. ? Ans. 60. 

23. Two travellers set out at the same time from Lon- 
don and York, whose distance is 197 miles ; one of them 
goes 14 mites a day, and the other 16 ; in what time will 
they meet? Ans. 6 days 13^ hours. 

24. There is a fish whose tail weighs 9tb., his head 
weighs as much as his tail and half his body, and his body 
weighs as much as his head and his tail ; what is the whole 
weight of the fish ? Ans. 72/6. 

25. It is required to divide the number 10 into three 
such parts, that, if the first be multiplied by 2, the se- 
cond by 3, and the third by 4, the three products shrill be 
all equal. Ans. 4^*3, o^\. and 2^*^ 

26. It is required to divide the number 36 into thr«e 
such parts, that i the first, i of the second, and } of the 
third, shall be all equal to each other. 

Ans. The parts are 8, 12, and 10 

27. A person lias two horses, and a saddle, which, of 
itself, is worth 50/. ; now, if the saddle be })ut oa the 
back of the first horse, it will make his vaiue double that 
of the second, and if it be put on the back of the second, 
it will make his value tripie that of the first : what is the 
value of each horse ? Ans. One 30/. and the other 40/. 

28. If A gives B 5s. of his money, b will have twice as 
much as the other has left ; and if b gives a 5s. of his mo- 
ney, A will have three times as much as the other has left : 
how much had each ? Ans. a 13s. and b lis. 

29. Whr-d two numbers are those whose difference, 
sum and product, are to each other as the numbers 2, 3, 
and 5, respv-ctirely ? Ans. 10 and 2 

30. A person in play lost a fourth of his money, and 
then won back 3s., after which he lost a third of what he 



128 QUADRATIC EQUATIONS. 

now had, and then won back 2s ; lastly, he lost a seventh 
of what he tlien had, and after this found he had but 12a. 
remaining ; what he had at first ? Ans. 2us. 

3i. A hare is 50 leaps before a greyhound and takes 
4 leaps to the greyhound's 3. but 2 of the greyliouud's 
leaps are as much as 3 of the hare's how many leaps 
must the greyhound take to catch the hare ? 

An?. 300. 

32. It is required to divide the number 90 into four 
suih parts, that if the first part be increased by 2, the se- 
eond diminished by 2, the third multipln.d by 2, and the 
fourth divided by 2, the suai, difference, product, and 
Quotient, shall be all equ;.i ? * 

^ Ass. The parts are 18, 22, 10, and 40 

33. There are three numbers whose differences are equal; 
(that is, the second exceeds the firJt as much as the third 
exceeds the second), and the first is to the third as 5 to 7 ; 
also the sum of the three numbers is ^24, what are tnose 
numbers' Ans. 90, 108, and 12G. 

34. A man and his \Vife usually drank out a casK ot boer 
in 12'davs, but when the man was from home it la-^led the 
woman 30 days ; how many days would the man alone be 
in drinking it / . . ^ ^ Ans 20 days 

35 A o'e! eral ranging his army m the torm ot a sulia 
square finds he has 284 men to spare, but increasing the 
side bv' one man, he wants 2a to fill up the square ; how 
many soldier^ had he ? . Ans. 24U00 

36. If A and b together can perform a piece ot work m 

8 dav« A and c together in 9 days, and b and c in 10 days, 

how many days will it take each person to perform the 

the same work alone ? , ^o - 

Ans. A 14ff days, b 17|^ and c 2:J- 

aUADRAllC EQlTATiONS. 

A auADRATic EQUATION, as before observed, is that 
jn which the unknown quantity is of two dimensions, or 
which rises to the second power ; and is either simple or 
compound. 



QUADRATIC EQUATIONS. 132 

A simple quadratic equation, is that which contains only 
the square, or second power, of the unknown quantity, as 

b , h 

ax^ = b, or X" =- ; where x=^-. 
a a 

A compound quadratic equation, is that which contains 
both the tirst and second power of the unknown quantity ; 
as 

a a 
In which case, it is to be observed, that erery equation 
of thi5 kind, having any real positive root, will fall under 
one or other of the three following forms 



.* X2 



-f flar=6 . . . where a: = — -±v'(— 4-^)« 



- * It may be observed, with respect to these forms, that 

1. Ih the case x2 -J. ax — 6:=o, where x= — ia -j. ^(JaE +b), or — ga— 
^ (la2 +6), the first value of ;r, must be positive, because \/{\a2 -f.6) is 
greater tlian y^/ias, or its equal la ; aud its second value will evidently be 
!iegati\e, because each of the terms, of which it is composed ii negative.^ 

2. In the casea;2 — ax — 6=0, where .x=^a + ^ ($a3 +6),or5a— ^/ {^a2 
-i-b), the first value of x, is manifestly positive, being the sum of two posi- 
tiTe terms: and the second value will be negative,, because ^(:Ja2 -f- 6) 
k greater than ia2 , or its equal \a. 1 /i 

3. in the case a2 — ax-f 6;=o, where a:-=^a-J- v'(in2 — b), or Jo — ^/(* 
a2 — b), both the values of x will be positive, when }a2 is greater than 6 ; 
for its first value, is tlien evidently positive, being composed of two positiv« 
terms; and its second value, wi"ll also be positive: because v'Ci*'^ — b) is- 
le?s than s/ iae , or its equal ^a. 

But if la2, in this crise be less than '. the solution of the proposed equa- 
tion is unpossible ; because tlic quantity 502 —6, under tlie radical, is the 
negative; and consequently y/ i\a2 —b) will be imaginarjs or of no assign- 
able value. 

4. It may be also further obser\ed, that there is a fourth case of tlie torn* 
x3-|-ar-J-fe=o, where a:=: — ^a-}-v'(J-a2-i-fe), or i-^a— -/(ja^— fc), 
the two values of x will be both negative, or both imaginary, accarding as 
\a2 is greater or less than b ; the imaginarj' roots, when they occur, being 
here of the forms — (n'-f c' v' — I) and — (a' — c ^ — 1). 

From which it follows, that if all the terms of a quadratic equation, when 
brought to the left hand side, be positive, its two roots will be both negative, 
or both imaginary ; and conversely, if each of the roots be negative or each 
inv.iginary, the signs ot all the terms will be positive. 

So that of all quadratic equations, which can have any real positive root, 
that of the third form, x2 — ax _ ir=o, is the only one, where the solutioa, 
for certain auaieral values of a aud 6, wiii become impossible. 



130 QUADRATIC EQUATIONS. 



^ 



2. X —ax=^b_. . '. where x=+-±v'(- — h^J 



a^ 



3. X- — flx= — 6 . . where x=-|--±v/i' -r — ^j- 

Or,il the second and last terms be taken either positive- 
ly or negatively, as they raay haj^pen to be, the general 
equation 

b c 

ax^ ±bx= die, or x" ±-x= ±- 

a a 

which comprehends all the three cases above mentioned, 

may be resolved by means of the following rule : 

RULE. 

Transpose all the terms that involve the unknown quan- 
tity to one side of the equation, «nd the known terms to 
the other ; observing to arrange them so that the term 
which contains the square of the unknown quantity may 
be positive, and stand first in the equation. 

Then, if this square has any coefficient prefixed to it, 
let all the rest of the terms be divided by it, and the equa- 
tion will be brought to one of the three forms above- 
mentioned. 

In which case, the value of the unknown quantity x is 
always equal to half the coefficient, or multiplier of x, in 
the second term of the equation, taken with a contrary 
sign, together with ± the square root of the square of this 
number and the known quantity that forms the absolute or 
third term, of the equation.* 



* This rule, which is rr.ore commodious in its practiLai application, tiian 
that usually given, is found, d upon t;:e s;»rTie principle ; being derived from 
the well kiiown propert}', that in any quadratic 

.12 ±nx^s: ±b if the square of half the coefficient a 
of the second term of the equation be added to each of its sides, so as to ren- 
der it of the form 

x2 +ar-}- Ja3 = ia3 jr 6 
that side which contains the unkno vn quantitv will then be a complete square ;' 
and, consequently, hy extracting the root of each side, we sha ll haye 



QUADRATIC EqUATIONS. 1^1 

Ao/c. All equations, which have the index of the un- 
known quantity, in one of tbetr terms, just double that of 
the other, are resolved like quadratics, by tjrst finding the 
value of the square root of thd? first terno. according to the 
method used in the above rule, and then taking such a root 
or power of the result, as is denoted by the reduced in- 
dex of the unknown quantity. 

Thus, if there be taken any general equation of this 
kind, as 

we shall have, by taking the square root of x^'", and ob- 
serving the latter part of the rule, 

And if the equation, which is to be resolved, be of the 
following form. 

rn 

we shall necessarily have, according to the same principle. 



EXAMPLES. 



1. Given a;=4-4x=l40,to find the value of x. 
Here x~-\r'ix=]40, by the question, 
Whence a:= — 2±^(4+l40), by the rule, 

Or, which is the same thing, x=— 2±^144, 



vhich is the same as the rule, taking a and 6 iii -f- or — as they may hap- 
>en to be. 

It may here, also, be observed, th.itthe ambiguous sign -4^, which denotes 
loth -f- and — , is prefixed to the radical part of the value of x iu every ex- 
iression of this kind, because the square root of any positive quantity, as as 
5 either -{-a or — a ; for (-f a) X (+<*), or ( — ")X( — «) are each = -j- 
s ; but the s<iuare root of a negative quantity, as — a^ , is imaginary, or un- 
ssif^nable, there being no quantity, either positive or negative, that, when 
lultiplied by itself, will give a negative product. 

To this we may also further add, that from the constant occurrence of the 
ouble sign before the radical part of tlie above expression, it necessarily fol- 
)ws, tiiat every qua Iratic equation must have two roofs; which are either 
oth real, or both imagmar) according to the nature of the questiua 



132 QUADRATIC EQUATIONS' 

Wherefore s= — 2+12=10, or — 2 — 12=-U, 
Where one of the values of x is positive and the other 
negative. 

2. Given a;^ — 12a:+30=3, to find the value of x. 
Here x^ — 12.t=3 — 30=-~27, by transposition, 
Whence a; =6 ± ^Z (36-27), by the rule, 

Or, which is the same thing, x==6±y/9y 
Therefore r=6+3=9, or =6-3=3, 
Where it appears that x has two positive values. 

3. Given 2:^2 -j-Ss:— 20=70, to find the value of x. 
Here 2a;2-f8x=70+ 20=90, by transposition, 

- And x2-}-4x=45, by dividing by 2, 
Whence a;= — 2 + ^^(4-1-46), by the rule, 
Or which is the same thing, a;=— 2±^49. 
Therefore x=— 2-f-7=5, or =-2-7=— 9, "" 
Where one of the values of x is positive and the oth"er 
negative. 

4. Given Sx^ — 3x-f-6=5i, to find the value of x. 

2 
Here 3x^ — 3x=5i — 6=— - by transposition. 

o 

2 
And x^ -x= — - by dividing by 3, 

1 12 

Whence x-=-±-^(^-~-'), by the rule, 

^ , , .2^1 1.1 

Or, by subtracting - from -, x= -±^—, 

1,^2111 
Therefore x=- + g = -, or =2-6=3^ 

In which case x has two positive values. 

5. Given -s2 x-j-20i=42| to find the value of x. 

Here - x^ x=42|— 20^=221 by transposition, 

2 1 

Ajod x2 — - a;=44^, by dividing by -, or multiplie< 

by 2, 

Whence we have «=5±\/(Q+44i), by the rule, 






CtUADRATIO EQUATION'S, 133 

1 , 1 400 

Or, by adding - and 44i together, x— - ±-v/""9~' 

Therefore a;=-+6f=7, or =-— 6|=-6i, 

Where one value of x is positive, and the other ne- 
gative. 

6. Given ax^-\-bx.=c, to find the value of a-. 

be . " 

Here x^-\ — x=- by dividing each side by c. 
a a '' 

b yb^ c. 

Whence, by the rule, x=-— ±v'(— 4-}, 

, , b , b^-\-4ac 

t)r, multiplying c and a by 4a,x=——±^ — ^^^ , 

Therefore i—~—+—v/(i2 4.4a<:). 

7. Given ax^ —bx-{-c=d, to find the value of x. 
Here ax^—bx=d — c, by transposition, 

And x^ — x= by dividing by a. 

b d'^c b^ 

Whence x——- + ^( -i--r^) ^y the rule, 

2a-~^ ^ a 4a2'' 

Or, mult? d-c & a by 4o, a; ~-^±a'\/{'^K^--(^)+^^) 

8. Given x*+ax^=b, to find the value of a:. 

Here x*-\-ax^=.b, by the question, / 

Orx2 = -|+^(^+Jj=-|+ly(a2+4fe), by the 

rule, 

a 1 

Whencex=42y'(— -4^-^(464-a^)) by extraction of 

roots. 

9 Giv«n-a:6 ar3= — -— , to find the value of x. 

2 4 32 , 

Here-a;^ ""I^^^""^' ^^ ^^^® question, 



134 (QUADRATIC EqUATIONl?. 

And x^ — -x='= — — , by multiplying by ?, 

•^ JO 

Whence a;3=-±v^(— — —),=- by the rule, 
4 It) lb' 4 

1 2 1 

And consequently ^^i'^T^ Vo~;5\/^- 

10. Given 2x3+3x3=2, to find the value of x 
s. 1 
Here Sx^-}- 3x3=2, by the question, 

3- 3 J- 

And x'' +o*^ = l> ^y dividing by 2, 



Wh 



¥ 3 , .^9 , X 3.5 \ 

Therefore x=r(-)3=-, or (— 2)3 = -8. 

11. Given x*— 12x3-|-44x3 -48x=9009(a), to find^the 
value of X. 

This equation may be expressed as follows, 
*(x2 -6x)3 4-8(x2— 6x)=a, 



* The method of expressing a biquadratic equation, by the form of a 
quadratic, as above, is found thus : 

Let a;4 ■^-20x2 -|-5a2a:2 -f4a3a;=rf, be the biquadratic, which I am t» 
resolve by a quadratic ; first, find the two first terms of the square root of the 
left hand side of the equation, the square of which take from said side ; then, 
if the remainder be divisible by the two terras of the root thus found ; tiie 
eijuation can be resolved by & quadratic equation. 

Example. 

354 .J. 2aa;3 iij. 5o2 a;3 + 4ffl3af'=d 
jr4 '^ 

2*3 +«m;)2<ix3 +5(j2 ,t2 {x2 + a%) 

2ax3 4- a2xz • 

I2 +aa; ) 4a2.r2 ^4a3a(4oa 
■ ' 4aa<|2 -J.4a3a; 

Hence (x2+(Wl:)3 +4o2(i2 4.aa:)=.r4 -f 2nr3 f 5«2i2 •)-4a3x, ami 
consequently (a;2 -f-oa)2 ■\. 4fl:?(i2 -[. oa)=i/; which can be resolvrd by a 
quadratic equation. £</»Y. 



QUADRATIC EQUATIONS. 135 

Whence x^ — 6x=— ^rtv^C^G+a), by the common rule. 
And, by a second operation, a;=3jf^(9- 4jf ^(16-{-a)) 
Therefore, by restoring the value of a, we have 

Or by extraction of roots, x=13, the Ans. 

EXAMPLES FOR PRACTICE.* 

1. Given x^ — 8x-rlO=l9, to find the value of x. 

Ads. a-=9. 

2. Given a;^ — x-40=170j to find the value of a:. 

Ans. x=\b. 

3. Given 3x3 +2a; --9=76, to find the value of x. 

Ans. x=5. 

4. Given ~x^ — ^a'-f-7t=8, to find the value of x. 

Ans. a.==H- 

5. Given -X -x/J=22i, to find the value of x. 

Ans. 1=49. 
6.t Given x+y<'(5x-}-10)=8, to find the value of x. 

Ans. x=3. 



* The unknown quantity in each of the following examples, as well as in 
those given above, has always two values, as appears from the common rule ; 
but the negative and imaginary roots being, in general, but seldom used in 
practical questions of this kind, are here suppressed. 

f Two values, of x are found, according to the process in resolving this 
question; that is, 18 and 3 ; but it appears, that 18 does not answer the con- 
dition of the equation, if we suppose that ^{bx-\-\(y) represents (he positive 
square root of 5a; -f- 10. The reason is, that 5x + 10 is the square of — >/{ax 
-f- 10) as well as of ,/ (Sx-^-lO) ; thus, by squaring both sides of the equa- 
tion ^/(Sa; -f-10) = 8 — a:, a new condition is introduced, and a new value of 
the unknown quantity corresponding to it, which had no place before. Here, 
10 is the value which corresponds to the supposition that x — ^{Bx-^IO 

:=8). 

Besides what is already said, it may be farther remarked, that one of the 
two positive roots of a quadratic equation of this kind, is also sometimes ex- 
cluded by a condition in the question. Thus, if it were required to find three 
numbers in geometrical proportion, such that the sum of the first and second 
shall be 10, and the diflerence of the second and third 24. 

It will be found, by putting .-rlO — x and 34 — x for the three numbers, that 
the values of .r, in the resulting equation x(34 a') = (10 .t)?, as found by 
tl.e usual rule, will be 25 and 2. 



JE36 <iUADRATIC EQUATIONS. 

7. Given ^(10+x)~V(iO+^)=2, to find the value 
^* Arm . y— — fi 

8. Given 2a.*-a2 + 96=99, to find the vahie of x. ' 

Ans. x=-^6. 

9. Given xo+SOa^a- 10=59, to find the value of x. 

Ans. x—%/3. 
IQ. Given Sx*"— 2x''+3=:l I, to find the value of .r. 

Ans. ar=V2. 
1 ^ 

?1. Given 6^«-3^.T=i-, to find the value of x. 

13 I 
Ans. 3- or-. 

12. Gi^en-Xv/(3+2xa)=-+-a;3,tofindthe value of 

^" Ans. ■x=-v'(^3+3y2). 



2 
13. Giv^n xy( — a?)=i — ^to find the value of x. 



^x 

2^ 



Ans. x=(l-|--^2)^ 



11. Given -v'(l-x3)=x3, to find the value of a:. 

X 



Ans. x=(-y5-^f . 



15. Givenxv^( — l)= v^(x2_fc2)^ to And the value 



of X. Ans. x;=ia-l — ^{^h^-\-a?) 



16. Given ^(l-fx-a-2)— 2(1 +x-x2)i^-- to find the 
value of :», Ans. x= — |--v'41. 



But as the sum of the liist and second of the required numbers is, by the 
question 10, it is manifest that 25 cannot be one of them ; thei-efore the onlv 
proper root of the sum in this cuse is '2 ; anil, consf-quentlv, the proportJonnTi 
sought ar« 2, 6, and 33. 



QUADRATIC EQUATIONS. 137 

17. Given y/(x )-Ty/{'i — -)=x,io finA the value 

of rt. Ans. x=\-\-^y/b. 

18. Given x^^-Sx^ +x"=6, to find the value of x. 

Ans. x=;X(i+iv/13). 

19. Given x« — 2x3-f-x=a, to find the value of a;. 



Ans.x=.x^.^|?±^(a+l>)| 



When there are more equations and unknown quantities 
than one, a single equation, involving only one of the un- 
known quantities, may sometimes be obtained, by the rules 
before laid down for the solution of simple equations 
and, in this case, one of the unknown quantities being de 
termined, the other? may be found, by substituting its value 
in the remaining equations. 

EXAMPLES; 

1. Given I ^.^ _2g S to find the values of x and y. 

no 

Here, from the second equation, we have y= — ; and 

784 
by substituting this in the first x^-\ — ~=-Q5, or x 4-- 65x2 

=—784. * 

. Whence, by the common rule before given, we have 

Or, by reducing the parts under the last 'radical, and 

extracting the rootx=±^(— ±— )=7, or —1, and cob- 

, 28 28 

sequently, y=-y, or ——=4 or - 7. 

Or the solution, in cases of this kind, may often be more 
readily obtained, by some of the artifices frequently made 
use of upon these occasions ; which can only be learned 
from experience : thus, taking as before, (1.) x^-\-y^~Z^5, 

n2 



Itja qUADKATiC EQUATIONS. 

(2 ) xu=28, we shall have, as in tbe former method, by 
multiplying by 2, 2xy=56, and, by adding this equation to 
the first, and subtracting it from the same, x2+2x2/+2/2 — 
121, and a-2— 2a:y+2/2=9. Whence, by extractmg the 
square roots of each of these last equations, there will 
arise, x+y=±U, and x-y=±3, and, consequently by 
adding and subtracting these we shall have 2x— ±14, or 
3=7 or -7, and y=4, or -4. It will also, sometimes 
facilitate the operation by substituting for one of the un- 
known quantities the product of the other, and a third un- 
known quantity which method maybe applied with advan- 
tage, whenever the sum of the dimensions of the unknown 
quantities is the same in every term of the equation. 

2. Given \ %X'4^=G0 \ ^^ ^"^ '^' ^^^^^ '^ " ^""^ ^- 

Here, agreeably to the above observation, let x=vy, 
then v'y^ +^i/2 =56, and vy' +2^= =60, whence, from the 

first of these equations, y'^^-^j-^ and from the second 



r= 



^Q . Therefore, by equating the right hand mem- 



^+^ 60 _ 66 

ber of these two expressions, we shall have^q-^— ^^jq;^, 

or C0x)2 -f 60r=^66r+ 1 12. And, by transposing 66v, and 

1 28 
airiding the result by 60, r= +Y^^= 15- ^^^^^ ^y ^^® 

_ 1 ^1 

common rule, for quadratics, we have^ 30-'^V900 

, 28\_. J_4_ll=l, And, consequently, by the for- 
'^T5>'"'" 30^30 3 

60 60 __ _ , 

per part of the process, 2/='=^^-JTq:^— ^^' °^ ^"^ ^ 

<18)=3v'2,andx=x'j/=^X3^2=4v/2. The work 

Mjav also be sometimes shortened, by substituting for the 
UBkBOvrn quantities, the sum or difference of two other 



QUADRATIC EQUATIONS. i39 

quantities ; which method may be used, when the un- 
known quantities, in each equation, are similarly involved 



a 



■^>r = 



12> 



3. Given \y x ^ to find the values of x and y. 

(x-i-y 

Here, according to the above observation, let there be 
assumed xzzz-\-v, and yz=.z — v. Then by adding these 
two equations together, we shall have x-{-y=2z=l^^ or 
2=6, also, since x=6-\-v, y=6 — v, and by the first equa- 
tion, x^-^y^ = \&xy, we shall obtain, by substitution, (6-j- 
v)3-f-(6— ■u)^ = 18(6+Tj)(6— 1>), or, by involving the two 
parts of the first member, and multiplying those of the 
second, 432-{-36xj2 =648 — IQv^ , whence, by transposition 

216 

64v3=:216 ; and by division, ^^^-rr^^"* 5 ^r t)=±:2. 

And, therefore, by the first assumption, and the first 
part of the process, we have x=^4-^'=6±2=8, pr 4, 
and ?/«=z — 1'=6±2=4, or 8. 



QUESTIONS PRODUCING QUADRATIC 
EQUATIONS. 

The methods of expressing the conditions of questions 
of this kind, and the consequent reduction of them, till 
they are brought to a quadratic equation, involving only 
one unknown quantity and its square, are the same as 
those already given for sample equations. 

]. To find two numbers such that their difference shall 
be 8, and their product 240. 

Let X equal the least number. 

Then will x + 8 = the greater. 

And x{x 4- 8)=j3 -f. 8a; = 240, by the question, 

Whence x =—4 + ^(16 -f- 240)= — 4 + V 236 by the 

common rule, before given. 

Therefore x = 16— 4 = 12, the less number, 

And x -j- 8 = 12 + 8 = 20, the greater* 



140 QUADRATIC EQUATIONS. 

2. It is required to divide the number 60 into two such 
parts, that their product shall be 8G4. 

Let X = the greater part, 
Then will 60-a; = the less, 
And x(60 _3:)=60x— x2 = 864, by the question, 
Or by changing the signs on both sides of the equation 

a;=-COx = — 864", 
Whence a;=30i:v^ (900-864) =30±y 36=30±6, by 

the rule, 
And'consequently a=30+6=36, or 30-6=24, the two 

parts sought. 

3. It is required to find two numbers such that their 
sum' shall be 10(a), and the sum of their squares 68(6). 

Let :B=the greater of the two numbers, 
Then will a— a=the less, 
And a-a-f (a-~a;)2=2:c3-2ax+a2=:6, by the questioa, 
Or 2a;2— 2ax=6— a2, by transposition, 
6 — a2 , ,. . . 
And x^ — ax= — - — , by division. 

Whence x=-±^y(-+-2-)=2^2v/(26-aO 

by the rule, 
And if 10 be put for a, and 58 for 5, we shall have 

a;„ l^_|.ly(116— 100)=7, the greater number, 

And lO-x=^-^V'(l ^^- 100)=3, the less. 

4. Having sold a piece of cloth for 24^, I gained aS; 
much per cent, as it cost me ; what was the price of the 

cloth ? 

Let x= pounds the cloth cost, 

Then will 24— x= the whole gain, 

But 100 : X : : X : 24— x, by the question. 

Or x2 = 100(24 -x) = 2400 -lOOx, 

Thatis, x2-f-100x=2400, , /. 

Whence x= - 50+^(2500+2400)= - 50+70=20 

by the rule, 



QUADRATIC EQUATIONS. 141 

And consequently 20/.= price of the cloth. 

5. A person bought a number of sheep for 80/., and if 
he had bought 4 more for the same money, he would have 
paid 1/. less for each ; how many did he buy ? 

Let X represent the number of sheep, 

80 
Then will — be the price of each, 

X 

And — ; — =price of each, if x-\-4 coflt 80/. 
a;-t-4 

^ 80 80 , - 

But — = — I — 1-1 5 by the question, 

X x-{-4 

^ 80.-r , , , . ,. . 

Or 80= -j-g, by multiplication. 

And 80x+320=80x+a:2-f 4a;. by the same, 
Or, by leaving out 80a; on each side, x2+4a;=320, 
Whence a:=-24-^(4+320)= -2-f 18, by the rule, 

And consequently x=\6, the number of sheep. 

6. It is required to find two numbers, such that their 
jum, product, and difference of their squares, shall be all 
equal to each other. 

Let a;=the greater number^ and y= the less. 

Then } *T^~^2__ a i^Y *^^ question. 



Hence 1= — - ■- =x— y, or x=y-\- 1, by 2d equation. 

And (2/+l)+y=J/(2/4-l) by 1st equation, 
Thatis, 2y-|-l=^='+2/; ory^—y^i, 

Whence 2/=-+^(I+l)=i+iy5, by the rule, 
Therefore j/=-+-^5= 1.6 180 . . . 

Anda;=:i/+l=-4--^5=2.6180 ... 

Where . . . denotes that the decimal does not end. 

7. It is required to find four numbers in arithmetical 
>rogression, such that the product of the two extremes 
hall be 45, and the product of the means 17. 



142 QUADRATIC EQUATIONS. 

Let x= Ifeast extreme, and y^^ coftimon difference, 
Then x, x-\-y, x-]-2y, and x-\-:iy, M'ill be the four number* 
Hence J '*(^+^^2/)=^'+3xj/=45 } bvtheaue' 

tion, 

And 23/3=77 — 43=32, by subtraction, 

32 
Or ^2=1 =ig by (jiyision, and 2/=y^ 16=4, 

Therefore x'-f-3a:j/=a;3-{-12a;=45, by the Ist equatior 
And consequentJy a = — 6+ v/C36-f 45)=— 6+9=3, b 

the rule. 
Whence the numbers are 3, 7, 11, and 15. 

8. It is required to find three numbers in geometricc 
progression, such that their sum shall be 14, and th 
sam of their squares 84. 

Let x, 1/, and z be the three numbers, 
Then x2=^y'^ , by the nature of proportion, 

And \ ^.t^+!|!;iL84 \ ^y '^^ ^"^^^^°°' 
Hence a— }-2'=14 — t/, by the second equation, 
And a;2+2za;+23=:]96_282/+t/3, by squaring bof 

sides. 
Or a;a 4-^2 +2t/2 = 196 — 282/4-2/^ by putting 'iy- for it 

equal 2a-z, 
That is x2 4-?/2 4-^2 = 1 96 _28y by subtraction, 
Or 196— 282/=84 by equality, 

Hence y=- — -^ — =4, by transposition and division, 

1 fi 
Again xz=j/2 = 16, or a;= — , by the 1st. equation, 

1 fi 

And x-\-y-\-2-=- f-4+z=14, by the 2d equation. 

Or 16+4z4-z2 = 14z, orr2_10z = — 16, 

Whence z=5±^(25— 16)=5±3=8, or 2 by the rule 

Therefore the three numbers are 2, 4, and 8. 

9. It is required to find two numbers, such that thei 
sum shall be 13(a), and the sum of their fourth power 
4721 (i). 



QUADRATIC EQUATIONS, M3 

' Let a;= the difference of the two numbers sought, 

Then will-a-f -x, or^^r-^= the greater number, 

2 2 2 . 

. , I 1 a — x ^, , 

And -a—-x, or — ^= '"^ '^sSj 

But ^^i:^+fcl^=&, by the question, 
16 16 

Or (a+a:)*+(a— x)*=165, by multiplication, 
)r 2a* 4-120^x3 -f2i* = 166, by involution and additioB, 
\nd x'»-i-6u3x-=86— rt*, by transposition and division, 
Whence x^=^^ Sa^ +^ {9a^+U -a*)=^-Sa^ -\- 

, ^B{a*+b), by the rule, 

^nd x=^-3a3+2^2(a4"+6), by extracting the root, 

Where, by substituting 13 for a, and 4721 for b. 

we shall have x=3, 

Therefore — ^=—=8, the greater number, 

=—=5, the less number, 

The sum of which is 13, and 8« +5* =4721 . 

to. Given the sum of two numbers equal s, and their 
product =/J, to find the sum of their squares, cubes, bi- 
juadrates, &c. 

Let X and y denote the two numbers ; then 

(!.) x+i/=s, (2.)a"2/=P- 
P'rom the square of the first of thei»e equations take 

wice the second, and we shall have 

(3.) x3-fy2=s3_2p=sum of the square*. 
\Iultiply this by the 1st equation, and the product will be 

Prom which subtract the product of the first and second 
jquaiioos, and there will remain 

(4.) x3 4-j/3=j3_3sp=:sum of the cubes. 
Multiply this likewise by the 1st, and the product will be 
r*-}-xy3-fx3y+i/*=s*— 3s2p ; from which subtract the 
product of the second and third equations, and there will 
remain 




H4 QUADRATIC E(^UATIONS. 

(5.) x* +2/^=5* — 4s'»p+2^2s=: sum of the biquadFates. 
And, again multiplying this by the 1st equation, and sub» 
tracting from the result the product of the second and ; 
fourth, we shall have 

(6.) x^-\-y^=s^ ■^bs^p-}-5sp'^=s\iTn of the fifth powers. 
And so on ; the expression for the sum of any powers in 

general being ar"* + j'" = «"" — fns'^-^p-\'^~^~^ s"-^p~ - 

?»(m-.4Xm-5) 3 _^ m(m^5){m-6){m-l) ^ | 

2-3 ^ ^ 2-3-4 P "' 

&c. 

Where it is evident that the series will terminate when 
the index of s becomes = o. 

^rESTIONS FOR PRACTICE. 

1. It is required to divide the number 40 into two such 
jarts, that the sum of their squares shall be 818. 

Ans. 23 and 1 7 

2. To find a number such, that if you subtract it from 
10, and then multiply the remainder by the number itself, 
the product shall be 21. Ans. 7 or 3 

3. It is required to divide the number 24 into two such 
parts, that their product shall be equal to 35 times their 
difference. Ans. 10 and 14 

4. It is required to divide a line, of 20 inches in length, 
into twd such parts that the rectangle of the whole and one 
of the parts shall be equal to the square of the other. 

Ans. 10^5-10 

5. k is required to divide the number 60 into two such 
parts, that their product shall be to the sum of their squares 
in the ratio of 2 to 5. Ans, 20 and 40 

6. It is required to divide the number 146 into such two 
parts, that the difference of their square roots shall be 6. 

Ans. 25 and 121 

7. What two numbers are those whose sum is 20 and 
their product 36 ? Ans. 2 and 18 

8. The sum of two numbers is 1|, and the sum of their 
reciprocals 3^ ; required the numbers. Ans. ^ and f 



GtUADRATIC EqUATIONb. 145 

9. 'J'he difference of two numbers is 15, and half their 
product is equal to the cube of the less number ; rec^uired 
the numbers. Ans'. 3 and 1« 

10. The difference of two numbers is 5, and the differ- 
ence of their cubes 1685 ; required the numbers. 

Ans. Sand 13 

11. A person bought cloth for 331, I5s. which he sold 
again at 21. 8s. per piece, and gained by the bargain as 
much as one piece cost him ; required the number of 
pieces. Ans. 15 

12. What two numbers are those, whose sum, multi- 
plied by the greater, is equal to 77, and whose difference, 
multiplied by the less, is equal to 12. Ans. 4 and 7 

13. A glazier bought as many sheep as cost him 60/., 
and after reserving 15 out of the number, sold the re- 
mainder for 64Z., and gained 2s. a head by them : how 
many sheep did he buy ? Ans. 75 

14. It is required to find two numbers, such that their 
product shall be equal to the difference of their squares, 
and the sum of their squares equal to the difference of 
their cubes. Ans. iy^S and i(5-f ^5) 

15. The difference of two numbers is 8, and the dif- 
ference of their fourth powers is 14560; required the 
numbers.* Ans. 3 and 11 



* In the solution of this qiiestipn, the process brings out the answer in the 
ibnn 13 -f-a.r=6 ; which is a cubic equation, and, therefore cannot be resolv- 
ed by the ordinary rules for quadratics; but we can sometimes reduce such 
equations to the form of a quadratic, and then resolve it according to the 
rules already given. 

Rule. 

Whenever, in the cubic eqsatioH of the form x5-^ax^h ; b can be divided 
into two factors, m and n so that, mP. •]- a=n, then the cubic can be resolved 
by a quadratic: thus, let the equation be a. 3 -]- 6a? = 20 ; now let 20 be di- 
vided into two factors 10 and 2 ; it is evident that 22 , or 4 added to 6, the 
coefficient of ar, is equal 10, tiie other factor. And, conseciuently, by multi- 
plying by X, the equation lx:comes .r'l -f-6t2=20.r, or :i'4 -^6x2 =10X2x'; 
tlien scjuaring 2r and adding this squaje to both sides we get xi -^10cr2 = 
('2z)2 -i- 10 (_2x), by completing the square «4-f-10.r2 -f. 25= (2:c)2 -|.10 
(2.t)-J.25,and extracting the square root aS •^S'ss^i+S, by transposition 

■5» ^Oi- lion/^/i v — ^ IP.?:* 



( 



a;2 =2i', hence a'=2. ' Edit. 

O 



146 QUADRATIC EQUATlOiN^. 

16. A company at a tavern had 8/. 15s. to pay for their 
reckoning; but, before the bill was settled, two of them 
went away ; in consequence of which thoae who remained 
had 10s. apiece more to pay than before -. how many were 
there in company ? Ans. 7 

17. A person ordered 7/. 4*. to be distributed among 
?onie poor people ; but, before the tnuney was divided, 
there came in, unexpectedly, two claimants more, by which 
means the former received a shilling a piece less than they 
would otherwise have done ; what was their number at 
first ? Ans. 16 persona 

18. It is required to find four numbers in geometrical 
progression such, that their sum shall be 15, and the sum 
of their squares 85. Ans. 1, 2, 4, and 8 

19. The sum of two numbers is 11, and the sum of 
their fifth powers is 17031 ; required the numbers ? 

Ans. 4 and 7 

20. It is required to find four numbers in arithmetical 
progression such, that their common difference shall be 4, 
and their continued product 176985. 

Ans 15, 19, 23, and 27 

21. Two detachments of foot being ordered to a station 
at the distance of 39 miles from their present quar- 
ters, begin their march at the same time ; but one party, 
by travelling ^ of a mile an hour faster than the other, 
arrive there an hour sooner ; required their rates oif 
marching ? Ans. 3J- and 3 miles per hour 

22. It is required to find two numbers such that the 
square of the first plus their product, shall be 140, and 
tlje square of the second minus their product 78. 

Ans. 7 and 13. 

23. It is required to find two numbers, such that their 
difference shall be 13 g^/^, and the difference of their cube 
roots I-}. Ans. 15f, and 2^|. 

24. It is required to find three numbers in arithmetical 
progression, such that the sum of their squares shall be 93 ; 
andif the first be multiplied by 3, the second by 4, and the 
third by 5, the sum of the products shall be 66. 

Ans. 2, 5, and 8. 

25. The sum of three numbers in harmonical propor- 



qUADRATlC EQUATIONS. 147 

tion is 191, and the product of the first and third is 4032 ; 
required the numbers. Ans. 72, G3, and 66. 

!26. It is required to find four numbers in arithmetical 
progression, stub that if they are increased by 2, 4, 8 and 
15 respectively, the sums shall be in geometrical progres- 
sion. Ans. 6, 8, 10 and 12. 

27. It is required to find two numbers, such, that if their 
difference be multiplied into their sum, the product will be 
5 ; but if the difierence of their squares be multiplied into 
the sum of their squares the product will be 65. 

Ans. 3 and 2 

2C. It is required to divide the number 10 into two such 
parts, that if the square root of the greater part be taken 
irom the greater part, the remainder shall be equal to the 
square root of the less part added to the less part. 

Ans. 5-fiyl9 and 5— |a/19. 

29. It is required to find two numbers, such that if their 
product be added to their sum it shall make 61, and if 
their sum be taken from the sum of their squares it shall 
kave 88. Ans. 7+^2 and 7—^2. 

30. It is required to find two numbers, such that their 
difference multiplied by the difference of their squares 
shall be 576, and their sum multiplied by the sum of their 
squares shall be 2336. Ans. 6 and 1 1. 

31. It is required to find three number? in continual 
proportion, whose sum shall be 20, and the sum of their 
squares 140. Ans. 61+^3-^5^, 64, and 65 -.^3-^-^. 

32. It is required to find two numbers whose product 
shall be 320, and the difference of their cubes to the cube 
of their differ'^nce, as 61 is to unity. Ans, 20 and 16. 

33. The sUiH of 700 dollars was divided among four 
persons, a, b, c and d, whose shares were in geometrical 
progression j and the difference between the greatest and 
least, was to the difference between the two means, as 37 
to 12, What were all the several shares ? 

Ans, 108, 144, 192, and 256 Dollars. 



148 CUBIC EQUATIONS. 



OF CUBIC EQUATIONS. 

A cubic equation is that in which the unknown quan- 
tity rises to three dimensions ; and like quadratics, or 
those of the higher orders, is either simple or compound. 

A simple cubic equation is of the form 

ax^=b, or x^=- ; where x=l/- 
a a 

A compound cubic equation is of the form 

a'3-f-ci'.T=6, x^-{-ax'^=^b, ot x^-]-ax' -^bx'=c, 
io each of which, the known quantities a, b, c, may be 
either -4- or — . 

Or, cither of the two latter of these equations may be 
reduced to the same form as the first, by taking away its 
second term ; which is done as follows : 

RULE. 

Take some new unknown quantity, and subjoin to it a 
third part of the coefficient of the second term of the 
equation with its sign changed ; then if tliis sum, or dif- 
ference as it may happen to be, be substituted for the 
original unknown quantity and its powers, in the pro- 
posed equation, there will arise an equation wanting its 
second term. 

JVote. The second term of any of the higher orders of 
equations may also be exterminated in a similar manner, 
by substituting for the unknown quantity some other un- 
known quantity, and the 4th, 5th, fcc. part of the co- 
efficient of its second term, with the sign changed, ac- 
cording as the equation is of the 4th, 5th, &c. power.* 



* Equations may be transformed into a variety of other new equations ; the- 
principal of which are as follows : 

1. The equation .t4 — 4:<.-3 — 19.T3-f.106x — 120=0, the roots of which are 
2, 3, 4, and — 5 ; hy chang-ins; the signs of the second and fourth terms, be- 
comes a;4-}-4w.T — 19a:2— 106-r— I20i=0, the roots of which are 5, — 2, — T... 
smd — 4 



CUBIC EQUATIONS. 14^ 



EXAMPLES. 

1. It is required to exterminate the second term of the 

-equation a;3-|-3aa;2 =6, or x^-{-3ax^ —b=0. 

3a 
Here x=z — ^=^ — <^t 

x^=z^ —3az^+3a2z—a^ 
Then ( 3ax- = -{-3az^ - Ba'^z+Sa^ 
—b= -b 



Whence z'^ — Sa^z+^a^ -6=0, 
Or z3—3a^z=b — 2a^, 
•in which equation the second power (^^), of the unknown 
quantity, is wanting. 

2. Let the equation a:' — 12a;2-f- 3a:==- 16, be tr^ne- 
formed into another, that shali want the second term. 

Here x=z-\-4, 

5(^+4)3=23 4-. l2^2_L48^+64 
_ 1 2(z-}-4)2 -— - 12z2 _ 9t-z- 1 92 
- +3(^+4) = +3-4-12 



Whence z'^ — 45z — 116= — IG 
Ov z^-^45z=\0O'^ 
which is an equatioB where z^ , or the second terra, it 
wanting, as before. 



2. The equation a; 3 4x2 — 10 r+ 8=0, is transformed, by assuming .r=:=j 
.. — 4, intoj'3 — ]!y2 +30y=0, or3/2— lly + 30=0; the roots of which are 
greater than those of the former b3' 4. 

3. The equation j:3 — Gx2 +9^- — 1=0, may be transformed into one which 
shall want the third term, by assuming or=t/-{-€, and in th-i resulting- equa- 
tion, let Sea — ^12e +9, or e2 — 4i;.-f-3=0, in which the values of e are 1 and 
3: then assume a;=i/+3, or y+1, and the resulthig equation will be 1/3 + 
b'j/C — 1^=0, an equation wimting the third terra. ■ 

4. The equation 6a;3— lla;3 -}. 6a— 1=0 by assumuig x=z-, may be tans- 

formed i-^toys — 6y2 +lly— 6=9; tlie roots of" which are the leciprocals of 

the former. y ' 

5. The equation 3x3^—137.3 + 14x^-16=01)7 assuming 0=3, into ^3— 

"2'j~ +42y-J-144=0, the roots of which are three times those of the fcime: 

S 2 ■ - ■ 



150 CUBIC ECiUATlONS. 

3. Let the equation x^ —6x^ = 10, be transformed into 
another that shall want the second tefm. 

Ans. y^ — 12?/-=26 

4. Let 1/3 — 132/24-81y=243, be transformed into an 
<>quation that shall wan-t the second term. 

Ans. x3-j-6x=88 

3 7 9 

5. Lei the equfition x^-r -a:2-j--x— 77:=:0, be trans- 

4 8 16 

formed into another, that shall want the second term. 

Ans. i/ = +|g^/=^ 

6. Let.the equation x* +8^3 — 5;c2 -f lOx — 4 = 0, be 
transformed into another, that shall want the second term, 

Ans. y^ -291/3-}- 94!/ -92=0. 

7. Let the ecjuation x* — Sx^-j-Sa-^ — 5x — 2=0, be trans- 
rormed into another that &ha!l want the third term. 

Ans. y*-{-y3 — 4?/- 2=0. 

8. Let the equation 3x^ — 2a:-f-l=0, be transformed 
into another, whose roots are the reciprocals of the former. 

Ans. 2/3 — 2^2_|_s = o. 

9. Let the equation x* —^x^-h^x- —:^x+-f\=0, be 
transformed into another, in which the coefficient of the 
highest term shall te unity, and the remaining terms inte- 
gers. Ans. y* -Sy^-\-\2y= — W9y'{-12=0. 



OF tHE SOLUTION OF CUBIC EQUATIONS 



RULE. 

Take away the second term of the equation when ne 
cessary, as directed in the preceding rule. Then, if the 
numeral coefficients of the given equation, or of that 
arising from the reduction above mentioned, be substituted 
for a and b in either of the following formulae, the result 
will give one of the roots, as required*. 



» If, instead of the rcfMlar method oi reducing a cubic equation of the 
gancral form. 



CUBIC EqUATIONS. ^ 151 



or 



Where it is to be observed, that when the coefficient a, 
of the second term of the above equation, is ne^tive, 

CL^ ft, 

— , as also-, in the formula, will be negative ; and if the 

absolute b be negative,- in the formula, will, also, be ne- 

b^ . . 

gative ; but — will be positive.* ^ 



s 3 -f-aa? + 63- + c=^0. 

to another, wantir.p; tjie second term, as pointed out in the preceding article, 
there be put, a;=i{^ — a), we shall haye, by substitution and reducdon, y^ 
+ (96— 3rt3 )y=dab-^27c — 2a 3 ; where, since the value ofy can be deter" 
mined, by either of the formula given in this rule, the value of a; will also be 
know'i, being a^=i {i/—a). And if 6=0, or the original equation be of the 
following,- form a-3 -f- oas-f c==0, the reduced equation will be yo— SaZjf— 
w_ 2a 3— 27c, where the value oty, being found as above, we shall have, as be- 
fore, *=1 (y—a), which formulT, it may be observed, are more convenient, 
in some cases, than those resulting fropn the preceding article ; as the coeffi- 
'itnts, (hus obtained, are aUvays integers; whereas by the former method 
thev are fiequently fractions. . /-, i 

* This method of solving cubic equations is usually ascribed to Cardan, 
a celebrated Italian analyst of the Jetii century; but the authors of it were 
.S.ipio Ferrcus, and Nicolas Tartalea, who discovered it about die same 
time, indoiiendcntly of each other, as is proved by Montucla, in his Histo- 
ire *.; JMuthemaiiqucs, Vol. I. p. 568, and more at large m Button's Mathe- 
matical Dklionary, Art. Algebra. 

The rule above given, which is similar to that of Cardan, may be demon- 
strated as foliovtS : 

Let the eCjTJatiftn, whose root is required, be a?3 -f ar=&. 
And assume »/ + 2=:t', and %i=^—n. 
'. Tl.pn, by subsiitutina; thos<- \ allies in the given equation, we shall have 

=-yT +.i3-.-jx(y+2)+ aX(ifc-'-*)=^. or 



152 • CUBIC EQUATIONS. 

It may, likewise be reraarked, that when the cquatic. 
is of the form 

a^ h- 

and -;;; is greater than -—, or Aa'^ greater than 276 , the 

solution of it cannot be obtained by the above rule ; as 
the question, in this instance, falls under what is usually 
called the Irreducible Case of"cubic equations.* 

And if, from the square of this last equation, there be taken 4 times the 
cube of the equation yz=—-^a, we shall have yG — 2y2z3 ^z6=b2 + 
_«,(j2, or 

But the sum of this equation and ys -j- 23=i, Is 2y2=Jj-}-^' (bz 4, 
■^%!a3) and their difl'erence is2z3=^b— v'Ciz -J-^-'yas) ; whence 2/c=^(i 
6+ v/(d/'3..^'^^«3)), and ^=y(i6-^/(i62Vj^a3)). 

From which it appears, that y+z, or its equal x, is = 
v'Ci^ + \/ ii^- +o\a3))-i-^ilb~^{ib2 -f 2\a3)), which is the theo- 
rem ; 

a . .,, , , a 

Or, smoe z is == — — , it will be y-j- s=y~—, or oc= 

rule. 

* It may here be farther observed as a remarkable circumstance in the 
history of this science, that the solution of the Irreducible Case above men- 
tioned, except by means of a table of sines, or by infinite saricj, has 
hitherto bafHed the united edbrts of the meST celebrated mathematicians in 
Europe ; although it is well known that all the three roots of the equation 
are, in this case, reaJ ; whereas, in tliosc that air resoh able bj the above 
formula, only one of the roots is real, so that in fact, the rule is only applica- 
ble to such cubics, as have two equal, or two impossible roots. 

The reason why the assumptions, irii.de in the note to the former part of 
this article with respect to the solution of the equation a- 3 — ax = 6, arc found 
to fail in the case in cjueslion (and it does not appear that any otiier can be 
adopted) is, that the two auxiliary equador.s 3y2= — a and y'J -i-zs^^b, 
which in this case, l>ecorae 3yz'= a, and 2/3 + z3 = 6, or y3s3 = 

— , andy3-j-«-'=6, cannot take place together; being inconsistent with 

each otlier. 

For the greatest product that can be formed of the t%Vo quar.titif ? y\ •^- zS 
is, ^^hen they are all equal to each other ; or since y3 -^J-aS^i, whe.T each 
of these z= Xb ; in whicli case tlieir product iss^lbz . 

a3 , «3 62 

But, as above shown y^z^=: — , by the question, tnerefore when ^ > -r 

the two conditions are incompatible vvith each other; and consequently the 
solution of the problem, upon that supposition, can only be obtained by iic- 
aginary quantities. 



CUBIC EQUATIONS. 153 



EXAMPLES. 

Given2x3 — 12x2+36a;=44, to find the value of x. 

He»3 x^ — 6x2 — 18*=22, by dividing by 2. 
.\nd, in order to exterminate the second term. 

Vntx=z-^-=.--z-[-2, 

Then -6(^+2)2= -^6z^ -~2iz--U =22 
18(^-1-2) = 18Z+36 



Whence z3+6z+20- 22, or 2='+6r=2, 
And, consequently, by substituting 6 foro, and 2 for b, in 
the first formula, we shall have, 

,2 4 , 216,. . .2 A , 216., 

i. y(14-^(l+8)) + y(l-v/(l+8))=:3/(,^.y9)4. 

Therefore x=z-f2==V4- 3/2+2 = 2+1.687401- 
1.269921 = 2.32748, the answer. 
2. Given x^ —6x— 12? to find the value of x. 
Here a being equal to -G, and b equal to 12, we shall 
have, by the formula, 

i-_^2.2435+.8967=3.1392 

2.2435 

Therefore x=3.1392, the answer. 
-: 3. Given .t3_2x=~4, to find the value of x. 

Here a being =-2, and 6=— 4, we shall have, by the 
formula. 

r=i/l~2+v/(4-|^)HV{-2_v'(4-^)l. or 



164^ CUBIC EQUATION'S, 

by reduction, 3/(-2+-i^v/3)- V("'+^^^3)== 

3/(-2+ 1 .9245) -3/(2+ 1 .9245) = 3/(_.0755 - 
3/3.9246= —.4226 - 1 .5773= - 1 .9999, or - 2 
Therefore x=— 2, the answer.* 
Note. When one of the roots of a cubic equation has 
been found, by the common formula as above, or in any 
other way, the other two roots may be determiued, as fol- 
lows : 

Let the known root be denoted by r, and put all the 
terms of the equation, when brought to the left hand side, 
=0 ; then if the equation, so formed, be divided by x±Lr, 
according as r is positive or negative, there will arise a 
quadratic equation, the roots of which will be the other 
two roots of the given cubic equation. 

4. Given x* — 15a:=4, to find the three roots, or values 
of X. 

Here x is readily found, by a few trials, to be equal to 
4, and therefore 

a; — 4)x3 — 16a;— 4(jc3 +4a;-|- 1 



a." 


— 1U,- 




4x3 — 15x 
4x2 -IGx 




x-4 

.T-4 



* When the root of the given equation is a whole number, this method only 
determines it by an approximation of 9s, in the decimal part, which suffici- 
ciently indicates the entire integer ; but in most instances of this kind, its 
▼alue may be more readily found, by a few trials, from the equation itself. 

Or if, as in the above example, the roots, or numeral values of 3/^—2 -{- 

-5-%/ 3), and — v^(2-i — a'^^J ^ determined according to the rule laid 
down in Surds, fJase 12, the rcsujt will be found equal to — 2 as it ought. 



CUBIC EQUATIONS. 651 

Wlicnce, according to the note above given, 
x3 4-'ii--fl = 0, or x2 4-4.r=— 1 ; 
the two roots of which quadratic are — -+ V ^ and — 2- 
y'S ; and coiisequently 

4,-24--x/3, and--2-v^3, 
are the three roots of the proposed equation. 
Or, putting c=— 15 and//=4, we shall have, 

and c=^> 2- At- 27) n^ S^~v^^'*-^^^^)H 

as will be found either by cubing 2-}-^— 1 and 2 —n/ — l. 
or by the rule given in case 12 surds. 

Whence a+c=2+ y' _ ] +2— y' — 1=4, 

I- 2+^3, 

2-^3 ; 
and consequently 4, —2 + ^3, and —'i-^t/'d are tht 
three roots of the equation, as before found. 



EXAMPLES FOR PRACTICE, 

1. Given x^-\-3x- — 6a;=8, to find the root of the equa- 
tion, or the value of x. Ans. a==2. 

2. Given x3+a;2== 500, to find the root of the equation, 
Dr the value of x. Ans. a-=7. 616789. 

3. Given x^ —3x2=5, to find the root of the equation. 
w the value of x. Ans. a:= 3. 103803. 

4. Given a;3--6«=6, to find the root of the equation, 
ar the value of a-. Ans. 3/4^3/2. 

5. Given a;3-|-9a-=6, to find the root of the equation, 
Dr the value of x. Ans. ^9~^3. 

6. Given x3 + 2x2 -2,%= 70, |o ^^^ the root of the 
equation, or the value of x. Ans. x= 5.134899. 

7. Given x^ — 17x2+54x=350, to find the root of the 
equation, or the value of x. Ans. x.=; 14.934068, 



156 CUBIC EQUATIONS. 

8. Given x^—Sxr=i', to find the three roots of the 
equation, or the three vahies of x. 

Ans. -2, 1+^/3, and \-^3. 

9. Given x^ — 5x^ +2 r= - 1 2, to find the three roots of 
the equation, or the three values of x. 

Ans. — 3, l+y'5, and l--v^5. 



OF THE 

SOLUTION OF CUBIC EQUATIONS 

BY 
CONVERGING SERIES. ' 

This method, which, in some" cases, will be found more 
convenient in practice than either of the former, consists 
in substituting the numeral parts of the given equation, in 
the place of the liberal, in one of the following general 
formulae, accorLi.g to which it may be found to belong, 
and then collecting as many terms of the series as are suf- 
ficient for determining the value of the unknown quantity, 
to the degree of exactness required.* 

]. x^-{-ax=^b.'\ 



* The method laid down in this article, of solving cubic equation, by- 
means of series was rirst given by Nicole, in the Jilemoirs of the Academy 
of Sciences, an. 1738, p. 99 ; and afterwards at greater length, by Claih.\ut 
m his Elemens d' A Igtbi-a. 

f With respect to the determination of the roots of cubic equations by 
means of series, let there be given, as above, the equation a- 3 -f-n.ar= i, \vhere 
tht; foot by transposing the terms of eaci> of the two branches of the cominoa 
formula, is 

l-l) J ; or, by putting, for the sake of greater simplicity, ^ {jb " + 2 V" ) 
:=c:.'!, and reducing the expression, a- =s¥ >\/^^"^2~- — V^^^~'5rM 



CUBIC EQ.UAT10NS. I£l7 

_ 26 ^i4-^"^r ^'^- > 2.5.8.11 

^'' (/{2{21b^+^la'))( • 6.'9^2r6M-4a3>' 6.9.12.15 
. 2763 ., 2.5. 8.11.14.17/ 276^ x ) 

V276=»+4a3>'^6.9.l2.l5.18.2lV2762+4a3y ) 

_ 2& ^ 2.5. 276= . 8.11 

^~3/(^2(2762+4a3))| «5.9U762+4a3^^"^12.15 
^ 2762 N 14.17 /- 2763 \ 20.33 

V2762+4a3y^'^18.21 V276M-4aV*^ 24.27 
~ / 2762 

In which case, as well as in all the following ones, a, b, 
c, &c. denote the terms immediately preceding those in 
which they are first found. 

2. x3 _rtx=±6, where 162 is supposed to he greater 
than ^'^ a3, or 276^ >4a3. 



.^ bi 2 .2762_4a3- 2.5.8 ,2762 ~4a3v, 

"" '^2^ 3.6*^ 2762 ; 3.6.9.12^ 276^ ^ 

^276^-40.3 > 

i.18^ 2762 ^ ^ 



2.5.8.11.14 ,2762 _4a3. 



3.6.9.12.16. 

Hence, extracting the roots of the right hand member of this equation, by 
ihe binomial theorem, there will arise V'^^'^o"''^ ^"^ ^^ 2~' ~ Tfi 

^2s^ ^3.6.9^s^ 3.6.9.12^ ^s'^ ^ 

2.5 / 6 \3 2.5.8 



y'^^""2r-'= '^27^ ~ 376*^^7'' "37679^ 2;^ -J 



6.9.12 



(i-)«_&c. 



And consequently, if tlie latter of these two series be taken from the former, 
ihe re'ult, by making the fii-st term of the remainder a multiplier, will give, 



264 t -2.5/ 6x3. „ ) 



6 X , 27^3 



wherev since s = x/ (56 ^ + 2 T " ' ). «^e shall have (— ) ^ = 276^+473 

rby,^ (-Jl^—y &c And ?^i= ^* ^ 

'2^^ ""^7624.403' ' • 6s 6.| ^(2(27t2 +4a3)) 

Whence, also, by substitution, we hare the above formula. 

* The root, as found by the common formula, when properly reduced, is 

P 



158 CUBIC EQUATIONS. 

bi 2 .2762— 4a3 5. 8 2762-4<t . ^ 

11.14 ^2762 _4a3, 17.20 2762- 4a% 
V 27A2 J 21.24^ 2763 J 



15.18^ 2762 

In which case the upper sign must be taken when b is 
positive, and the under sign when it is negative ; and the 
same for the first root in the two following cases. 

, 3. x^~ ax=±b, 
where ^b^ is supposed to be less than a^a', or 21 h'^ /L 

4a' 

_' ^ bi 2,4a3— 2762. ^2.5.8 . 40.3-2762 2 

:r=±2^-|l + g^( ^^1 >'~3.6.9.12^ 2762 -^ 

^._8JI1 J4_ 4a3-_276S3 _^^ K 
^3.6.9.12.15.18^ 2762 V ^ 



«3 



)) ^ . Or, putting, as in the last case ^v" (i 6^— 2T« ^ )' °'" '^= ^^^^^ 
^7h2-Aa3^l^^^ we shall have ^ = ± 3/| | ^(1 +*) + V(^^-*) } 



'^ 276 



Whence, extracting: the roots of the right hand member of this equation, 

, 2 o . 2.5 3 2.5.8 4, ^ 

theremllarise3/(l+5)-=l + 3*-3;^«- -+3:6:9* -5:6:912* + *'• 

, 2. 2 2.5 3 2.5.8 4 
3/(l_s)==l-i.-— .'^-g^g^s -3:6:9:12* -'^'^• 

And, consequently, by adding the two series together, and taking the first 

be ^2 

term of the result as a multiplier, we shall have a:= ±2 ^- |1— Jg* 

^5^ _2A8,lJ^^e_^ ) Or, by substituting (^f^) 

3.6.9.12 3.6.9.12.15.18 i ' "' ^ ^ 2762 

for its equal s, we get the above expiession. 

* This expression is obtained from the last series, by barely changing the 
sions of tlie numerator and denominator in each of its terms; which does not 

alter their value. . , , 

Hence, in order to determine the other two roots of the equation, let that 

above found, or its equivalent expression ^ H6-fv^(| 6^— ^'ja ') > 
Then, according to the formula that has bceo before given for these roots, 



\ 



CUBIC EQUATIONS. 159 

'•"iS.lG^ 2762 ^' 21.24^ 276^ > 

which series answers to the irreducible case, and must be 

used when 2a^ is less than 2762, 

And if the root thus found be put =r, the other two 
roots may be expressed as follows : 

•"—- i-2~ 9y2p" ( 6.9^ 2762 ^"^6.9.12.15 
.4a3_2762 _M:^iiljl4J7_,4a2--2762 ^^ ) 

^""276^ > "3.6.9.12.15.18.21^ 276= J •^' 

Or, , 



_—. r '•--3 

in the former part of the present article, we shall have x \- -± — - — 

j3/(46 + x/(i62-3Vo='))-»/(i6— v/(i^-2Ttt3))S • Or, putting 
-x/Ci^^— 1T«»"^) = *' ^^ reducing the expression, a:=+-j:: 
v4.t')3_%/— 3 < J _j_^^ — 3/(1—5) I . "V^Tience, extracting the cube roots 

of the right hand member of this equation, there will arise 

, 2 o 2.5 T 2.5.8 . 
3/(l + .) = l+is-3-r+3-ggS^-3-^^-s*4-&c. 

, 2 a 2.5 3 2.5.8 4 . 
^(l_,) = l_^.__s __-5 -3-;^;9-f2* -&c. 

And, consequently, by taking the latter of these series from the former, and 
making the first term of the remainder a multiplier, we shall have a;=ip 

r^ s({b)^,/-3 C 2.5 , 2.5.8.11 4 2.5.8.11.14.17 e^. } 
2 — 3 r+6.9 ^6.9.12.15* +6.9.12.15.18 21* ^'^^ 5 

2 , ■> 1 t / 2762 — 4a3 \, 2 2762 — 4a3 
But since 5=-^(.i62__V „3) ^ (^ ___ j_^ , =, ____=_ 

l^izl^ii, ,4 = (1^^±=?I^) ^ &c., and alg6 3/J- 6 X V-3 = t^ J- 
2762 ' — V 2762 > ' ' V 3 ^ 3"^ V a 

v/— 3 ,/2762 — 4a3x ,^,,, /4a3— 2762\ 

fcX V--/^-276i-^=3yi^X ^(— 9-,— ) = 

— ^^ i, if these values be substituted for their equals, in the last se- 

93/262 ' 

lies, the result will give the above expressions, for the two remaining roots of 
the equation. 



160 CUBIC EQUATIONS. 

r V(4a3~2762) f 2 . 5 Aa3—21b . S.ll 

^~^2— 9"^7263 J ^~6T'9^ i762 >^^ "^12.15 
.4a3 — 27i3N 14.17.4a3 — 2762, 20.23 . 4a3 — 276«. 
^""2762 Is.Tl^ 27P -/'^''^24.27^' 27A"2 ') 

Where — ir, or +i>", mast be taken according as 5 is 
positive or negative ; and the double signs ± must be 
.considered as + in one case, and — in the other, as usual. 

4. x^ — ax=±6, 
where ^b^ , is still supposed to be less than ^V^^* ^^ ^Tfi^ 
A4a3. 

_ 26 i 2.5/ 21b^ \ 

^~ — ^3/(2(4a3-2763))^ ^~6^\4a3-276='y'^ 
2.5.8.11 ^ 2762 ^2 2.5.8.11.14.17 ^ '2762 3 
6.9.12.1oV4a3 — 2762/ "" 6.9.12.15.I8.2lUa3-2762>' 

+ &c J . * Or, 



* By transposing the terms of the common formula, as in the first case, vre 
Sliall have x= 3/ j V (i*2 _^V« +^ b\-l/ \<^(lb ^ — aV « ' )— 
i 6? . Or, by putting, for the sake of simplicity, as before, V {iba — sTf 

n^)z=s, and reducing the equation ac=3/s | 3^(l +2P~VV~~2s ^ » 

AVhence, extracting the roots of the right hand member, as in the formed 
instances, , „ , , ^ ,- „ i 

f. bs ^ x(l>\ ^(b^i 2.5/6N3 2.5.8 
+ &c. 3/^^~27^=^~^^27''~8:6'^^^ "3X9 '^27'' ~3.6.9.12 

And, consequently, by taking the latter of these series from the former, 
and makini' ihe first term of the result a multiplier, we shall have 

2bsi 1,^2.5/6 -.a, 2.5.8.11 ,6 y , J:5^HilI- (i- V 
.^___ ^i + _(^_; +___i^_; +69 12.15.18.21^2.^ , 

> / b \^ 276a 

+ &C. I . But since j=v'(*6^-^t « ). we shall have (,— ; =276i^a3 



CUBIC EQUATIONS. 161 

26 ( _2^^ 27fc2 N^ 8.11 

''*''"" ■^3/(2(403 -2762)) i 6.9\Aa^-^1b^y 12.15 
j^ 2762 . 14.17. 2762 20^3^ 276^ 

V4a3-2762/'^'' 'iS^aTMa^ -^76- 3^'^24.27^4a»--2762 ^^ 
»-&.€., which series also answers to the irreducible case, 



= ?!^^,rA) 4 ^ (_^^i__) ^&c., and '-^=11: = 

4a3— 2762'^ ^s"^ ^4o 3^-2762'^ 6 s 6sf 

26 26 



Whence, if these values be substituted for their equals in the last series, 
there will arise the above expression for the first root of the equation. And, 
if we put the root thus found, or its equivalent expression 

we shall have, according to the formula before given for the other two roots, 

— -g-fc) j . Or, taking, as before, v'(i6^ — 2V «')=«» and simplifying 

the result, ..^ + -r-H*-i^^ j 3/(l+ ^) +3/(l-^) | . - 

Whence, by extracting the i-oots of the right hand side of this equation, 
there will arise 

^6 17 ^ ^ ^ r ^ V 2.5 ( b \3 2.3.8 / 6 y 

b\ . 1^6 \ 2 ^ 6 ^2 2.5^6 >^3 2.5.8 

3.6.9.12 



+ «c. 3,( ,_^)=.,_K^)-3T5(^)^-3^,(i)=-, 



i^y-"- 



2s 

And, consequently, if the latter of these series tie added to the former, we 
shall have, by making the first term of the result a multiplier, 

__»• 1 y ^ U 2/6 >,2_ 2.5.8 (b X4 2.5.8.11.14 

'^— "^a — *^^ r~ 3:6^25'' 3.6.9.12^2*^ 3.6.9.12.15JS 

{ — )^—&c. But since s=v'(Jfc2— 2V «^)=V—-j27 — -'"'^''^ sha» 

f b \s 2763 2762 o. ■, , 

alsohave(-)2=^^^_^^_=_^^^-2^^-&c. and consequently,. 

Hence, if these values be substituted for their equals in the above series, 
the result will give th« abeve expressions for the two remaining roots of tbe 
«quatioE. 

? 2 



162 CUBIC EqUATIONS. 

and must be used when 2a ^ ig greater than 27&3. And if 
the root thus found, be put =r, as before, the other two 

r 4a 3 — 2762 c 
roots may be expressed thus: a;=rp-±^ \ 1 + 

_2_. 276- . 2.5.8 276= ..a 2.58.11.14 

sTdMaS- 27^2 ''""3.C.9.12V4a3 -2762'' "^3.6.9.12.15.18 

r. 4a3 — 2763t _ 2 , 276^ . 6.8 



x=q:-±e/- 



C 2, 2763 5.8 

\ "^sTeMa'— 2763>'* 9.1i 



2 ^ 4 ^ 3.6 ^4a'— 2763 >' 9.12 

. '21b^ . 11.14. 276 2 . 17.20. 276^ ^ 
Ma3-2762'' ^ "^ 15. 18Ma3 — 2762-' *^~21.24Ma3 —2763'' 

Where the signs are to be taken as in the latter part of 
the preceding case. 

EXAMPLES. 

1. Given rr3+6a;=2. to find the real value of x. 

Here a=^&, and 6=2, whence 

2762 27X4 11 

= P =- ; and 



2762+4a3 27X4+4X216 1+8 9 
26 4 

^(2(2762 + 4a''))~3/(2(4X27+4X216)) 



4 4 



23/81 3/648 _ . 

--.^^=y——. Consequently, 



23/(27 + 8X2T) 63/9~ 27 ~ 27 

\y formula 1 , we shall have 

1 1.0000000 (a) 

2 5 1 

~X- A .0205761 (b) 

o.y 9 ^ 
8.11 1 

12.15^9^ -0011177(0) 

14.17 1 , , 

•^,X-^c .0000782 (n) 

20.23 1 , , 

S,;r,X8» .0<»0062(.) 



26.29 1 
30.33 9^ 



CUBIC EQUATIONS. 163 

.0000005 (f) 



Log. 3.0217787 
Log. 1/64B 
Colog. 27 

No. 3274801 



1.0217787 

0.0093670 
0.9371916 
8.5686362 

— 1.5151848 



Therefore x= .3274801 

2. Given x3 — 9a:= 12 to find the real value of or. 

Here a-=9 and 6=12 ; 

12 ^ ,2763_4a3 27X144-4X272 
whence 23y_=23/6 and-^^^^= ^^-^^ 

_144~108__36 _1 

Ui "T44~4' 

Consequently by formula 2 we shall have 
1 



2 1,. 
-376^4^^^) 

-971-2 '^SW 
11 14 1, ^ 

.l!:?2xl(o) 

21.24 4^ ' 
23.26 , , 

29. Si- 1, . 
•33:^^4^^^ 

Sum 

Comp. 



1.0000000 (a 
— 0277778 (b 

-.0025720 (c 

- .0003667 (d 
—.0000619 (e 
—.0000114 (f 

- .0000022 (g 

-.0307920 
.9652080 



164 



CUBIC EQUATIONS. 



Log. 969208 

Log. 2%/e or Log. 3/48 



—1.9864137 
0.5604137 



No. 3.522334 ,5468274 

therefore a;=3. 522334 . 
3. Given x' — 12a:=15 to find the three values of x, 
Here 0=12 and 6=15 ; 
b 15 ,4a3— 2762 

whence 23/-= 23/— =3/60 and- 



2 ^2 
4.12 -27.152 256-225 31 



276^ 



27.152 225 225 

Consequently by foroiula 3, we shall have 
1 
2 31 

■^3~:6 ^225^ 
5. 8 31 



— T^r:. X 



9.12 225 
. 11.14 31 

"^15.18 226^ 
17.20 31 

"21.24 223 ^ 

, 23.26_ 31 

H X — E 

^27.30 225 

Sum of + Terms 
Sum of — Terms 

Difference 



Log. 1.0146837 
Log. 3/60 

No. S. 97 1362 



+ 1,9000000 (a) 


+0.0153086 (b) 


*-0.00078!2 (c) 


+0.0000614 (d) 


-0.0000057 (e) 


+0.0000006 (f) 


+ 1.0153706 
- .0007869 


1.0145837 


.0062880 
,5927171 


.69900.\l 



Therefore the affirmative value of x or first rootj r: 

3.97 1 962, 



CUBIC EQUATIONS. 166 

A i ■v/(^« ^~2762) __ v/83 7_ ^(9X93) _ V93 
'^ ' 93/262 93/4Z0 9^450 ~33/450 ' 



and 



93/262 
4a3_2762 31 



2762 



225 



+ 1 
_2^ 31 

679^225"*" 
^ 8.11 31 

"^1X16^225° 
14.17 31 

18.21 ^225 *^ 

20.23 31 

-I X D 

^24.27 223 

26.29 31 
X: 



30.33 225 

Sum 

Log. 9760683 
Log. ^^93 
Colog. 3/4.50 
Colog. 3 

No. .4099445 



Also- 



r 

Last No. 



93/450 
Hence, 



9-^450 33/450 

1.0000000 (a) 
—.0255144 (e) 

+.0017186 (e) 

— 0001490 (d) 

+0000145 (e) 

—0000014 (f) 



.9760683 

— 1.9894802 
0.9842415 
9.1155958 
9.5228787 

— 1.6121964 

-1.9859810 
-t-0.4099445 



Resnlt —1.6760365 

Or -2.3959256 

Whence the three roots or ralues of x are 3.971962,— 
1.5760365 and -2.396925. 
4. Given x3— 6a:=2 to find the three Talues of ar. 
-26 —4 



Here 



V(2(4a3 - 276- )) 3/(2(4.6 3 - 27.4)) 



166 



CUBIC EQUATIONS. 
_ -4_ -2_ 2^49 



and 



27&2 



33/(2(4.8-4)) 63/7" SS/? 21 4a-'-2-;6a 

4.2 7 _ 1 _1 
4.63— 27.4~8^l'~7* 

Hence by formula 4, we shall have 
1 



2.6 1 

""6:9^7* 
8.11 1 

12.15^7° 
14.17 1 



18.21^7*" 
20.23 1 

"^24.^^7'' 
26.29 1 

,X- E 



30.33 7 



Log. .9752414 
Log. 2 
L. 3.49 
Colog. 21 

No. 339870 



1.000(^*000 (a) 
-.0264660 (b) 

+ .0018476 (c) 

— .0001662 (d) 

+ .0000168 (e) 

-.0000018 (f) 

Sum +.9762414 



— 1.9891120 
0.3010300 
0.6633987 
8.6777807 

— 1.5313214 



Therefore one of the negative roots or values of x, is 
— .339870= -r. 

, . 4a3_27fc3 4.63—27.4 

Agam V =V =«/(63- 27) = 



«/189 and 



4 

27^3 



1 



4a3_-2762 7 



Hence, 



. 2 1 
-^3X^7^ 



1.0000000 (a) 
+0.0168730 (b) 



5. 8 1 

, 11.14 1 

17.20 I 

X- D 

21.24 7 

■^27:30^7'' 



CUBIC EQUATIONS. 167 

.0008398 (c) 
+ .0000684 (d) 
-.0000066 (e) 
+.0000002 (r) 



Sum 1.0150952 



Log. 1.0160952 
Log. «/189 

No. 2.431741 



Therefore - 
2 

Last number 



.0065070 
.3794103 

.3869173 

+.169936 
±2.431741 



Result ■ +2.601676 

Or —2261806 

And consequently +2. 601676, -2.261806,and — .33887© 
are the three roots required, 

EXAMPLES FOR PRACTICE. 

1. Given a;3_|_9.T; = 30, to find the root of-'the equa= 
tion, or the value of x. Ans. a. =2. 180849. 

2. Given x^ — 2 x = 5, to find the root of the equation, 
or the value of a;. Ans. x — 2.0945515. 

3. Given x^— 3a; = 3, to find the root of the equation, 
or the value of x. Ans. 2.103803. 

4. Given .r^ — 27 x == 36, to find the three roots or va- 
lues of .r. Ans. 5.765722, — 4.320684, and — 1.445038. 

5. Given x3-48.i;2 =_ 200, to find the root of the 
equation, or the value of x. Ans. 47.9128. 

6. Given x^ —22a; = 24, to find the rootof the equation, 
>r the value of X. . Ans. 5.162277. 



i^ BIQUADRATIC EQUATIONS. 



OF BIQUADRATIC EQUATIONS. 

A biquadratic equation, as before observed, is one that 
rises to the fourth power, or which is of the general form 
x*-{-ax^-}-bx'^-i-cx-{-d=0. 

The root of which may be determined by means of the 
following formula; substituting the numbers of the given! 
equation, with their proper signs, in the places of the i 
literal coefficients a, b, c, d. 

ROLE I.* 

Find the value of 2 in the cubic equation ^^-j-^'ac — 

the former rules ; and let the root, thus determined, be 
denoted by r. Then find the two values of a- in each of 
the following quadratic equations. 



* This metJiod is that given by Simpson, p. 150 of his Algebra, whicS 
consists in supposing the given biquadratic to be formed by taking the diffe' 
rence of two complete squares, being the same in principle as that of Fer 
rati. 

Thus, let the proposed equation be of the form x4 -f-oa:3 +6a.2 + ca-4-^ 
=^ (l),Wvrhg ail its terms complete; and assume (a;2 -^ ^ax^2>)2 — [qx 
4-r)3 =a'4-4-oa;3 +6a-2 -J- cr -J- d. 

Then, if x3 -J. -kax 4- P and qx + r be actually involved, we shall have 

— 92 I — 2qr 
And, consequently, by equating the homologous terms, there will arise 
' 1. 2iJ + ja2— g2=6 
i 2. op — 2qr=:C 

3. p3—r2 =d 
wliere, since the product of the first and last of the absolute terms of the: 
equations is evidently equal to j of the square of the second, we shall havi 

^P'^+(ia2—b)p2—2dp-^d(jaa~b) = i(a2pa—2acp+c2). 
Or. by bringing the unknown quantities to the left hand side, and the known 
to me light, and then dfvi<}ing by 2. 



2p + ^a2 — b = q2 
ap — c =2qr 

p2 — d szr2 



: 



Iiiq.UADRATlC EQUATIONS. 169 

6 ^ ^ 

^ S 

and they will be the four roots of the biquadratic required, 

EXAMPLES. 

1. Given the equation a;«~l Oa;^ -{-35x2— 50x+24=«, 
lO tind its roots. 

Here a=-10, 6=35, c=— 50, and/i=24 ; 
Whence, by substituting these numbers in the cubic equa- 
tion 



From which last equation p can be determined by the rules before given 
i"or cubics. 
And since, from the preceding equations, it appears that 

q=./(2p + Xa2—b) and r=%^— , or x^(p2—d), 
* 2q 

it is evident that the several values of a; can be obtained from the quantities 

thus foMnd. 

For, because r4 -f aa:3 -j- 5x2 -^cx-^d, or its equal (x2 ■^. 2ax-{.p)3 — 
(qx-\-r)2^^, it is plain that (xs 4--^ax4-p)2=(qx^r)2 . And, there- 
fore, by extracting the roots of each side of this equation, there will arise 
x2 -L. Sao. -^p^qx + r ; or a;2 -f (ia — q)i=r — p. 

Whence, by substituting the above values of p, q, and r, for their equals, 

and transposing the terms, we shall have a:2 -^ ? lal+I ^ (2p -f- ^a2 — 6) > 

X -^p + i/ (jo 2 — d)=0. for the case where up — c is positive ; and 

a;2 + jianPV (2?-f |a2— 5) J x^p±s^(p2—d)=^, 

for the case where op — c is negative ; which two quadratics give the four 
roots of the proposed equation. 

And by putting ^=« f- — , in the reducing equation (2), in order to destroy 

:t3 second term,, the several steps of the investigation may be made to agree 
'•Titli the exptesiions given in the above rule . 



170 BIQUADRATIC EQUATIONS. 

we shall have the foUowins reduced equation, 

2_13 35 

^ "~T2"~"l08' 
which being resolved, according to the rule before laid 
down for that purpose, gives 

But, by the rule for binomial snrds, giveu in the former 
part of the work, 

y(35-j-18^-3)=|+^v'-3, andy(35-lBv/-3) 

=7-1^-3; - 

H7 1 7 1 > 7 

Wherefore .=-)--f2v^-3+---^-3|==-. 

And if this number be substituted for r, — 10 for a, 35 for 
b, and 24 for d, in the ttvo quadratic equations, 

=t=+(^<'-v')i°'+-('--50()^=-('-+g*)— /)('• 

they will become, after reducing them to their most sim- 
ple terms, 

x2_3x=— 2, and x" — lx= — 12 : 

3 13 1 

from the first of wliich :t=-±w'-=-±-= 2 or 1, and 

7 17 1 

from the second x=-r±^-=-±-=4 or 3 ; 

2 4 2 2 

Whence the four roots of the given equation are 1, 2, 3, 
and 4. 

Or, when its second term is token away, it will be ef 



'• wSucb is caet'slfraTs tut T&iMCir-i : isiA ia tizat £35^ ss 



'^ - . ,- 1 , 

J £sz3i r-'Ki i3)t tw» vafaKs <gf X, -1. T - ^ : £he iiafiixvjiig 
1 If 

aoid t^er vr¥ be the iv&r wte^ e£ t^ 



- ^ft. 

Tfaea, simoE ihes a; ■ie p^er out, bjs asiii =fti fees 

-- J '3 ■' ■:- lus Team* or" Ais -fats: st^HEHHi, 

Gt, r — ■ » .- 5-4j!=«-,^^ , J — r=^— - w =.t. 

WhaBcf, sjjEiErsciiinr Ac atpacre sa ■toe jteri af Hhese ftoBi -dioi a Ak ae- 
CuuaL, and UiOaB diBii£riB£^ ■&::' sade< of 'doe iiffiis^kKL. '<i^^ sioiX beiwe 

. :>f 'St jj mBFr be sonnE 'Ebt ibe mk befioie :;!»«» £r cntaK sass- 

, - I - 

r^ta.-:., E..5-V saiTf s^f=» *-f^, wu.*— f= — , ttaBR -wSL x^ia^ tt aii.- 



x- 



172 BIQUADRATIC EQUATIONS. 

Or the four roots of the given equation, in this last 
case, will be as follows : 

2. Given a;* + 12a;-- 17=0, to find thafour roots of the 
equation.. 

Here fl=0, 6=0, c= 12, and tZ=- 17 ; 

Whence, by substituting these numbers in the cubic 
equation 

we shall have, after simplyfying the results, 

23-j-i7z— 18, 
Where it is evident, by inspection, that z=l. 

And if this number be substituted for r, for b, and — 
17 for d in the two quadratic equations in the above rule, 
iheir solution will give 

.^.-^-fV2+v/C-i-x/18)=+iy2+^(-^-3^2) 
a:==4.|v2_^(-i-v/18)=+iv/2-y(-^-3y2) 
Which are the four roots of the proposed equation ; the 
two first being real, and the two last imaginary. 



11 b 1,1 b 

where p beiiig known, s and q are likewise known. 

And, consequently, by e>ttiacling the roots of the two assumed quadratics 
x2 +j9.r 4-9=0, and a;3 -j.r.t-f-s = 0, or its equal .c— ^x+Sg=0, we shall 

which expressions, when taken in -f ahd — , give the iour roots of the pro- 
posed biquadratic, as was required. 



BiqUADRATIC EQUATICNS. 173 



RULE* III. 

The roots of any biquadratic equation of the forms x* -\- 
<(,r-+6x+c=0, may also be determined by the following 
general formulas first given by Eulkr ; which are remark- 
able for their elegance and simplicity. 



~ * This method, which differs considerably from either of tlie former, con- 
sists in supposing the root of the given equation, 

a;4 4.na'3 ^bx + 0=0(1), 
to be of the following' trinomial surd form 

a = v'f + v^ ? + \/ »■ ;. 
where p, q, ?•, denote the roots of the cubic e<iaation, 

of whie h the coefficients J', g, and the absolute term h, are the unkaown 
quAntilies tliat are to be determined. 

Then, agreeably to the theory of equations Fjefore given, we shall have/* 
-f- ij -f. r.^^—f\ pq -f"/"'4" 9^"^^^ i pqr=h. And, by squaring et-ch side of 
the formula expressing the valu<», of x, 

a: 2 =/y -+- ,7 + }■ i- 2 v^ ;>9 -f- 2 v*;) >• -F 2 v/ 9 '•• 

Or, by subslituting/for its equal — (p-j-i} •j-r)t and bringing the term, 
so obtained, to the otlier side of the equation 

x2+f=2Ypq-^2^^pr+2^qr. 

Also, by again squanng-each side f,f this last expression, we shall have as* 
■^2fx2-j^2=Apq^^4pr-\-iqr-^ 8x/p2 qr-\- S y/ q2pr + 8^ r2pq. 

Or substituting -ig for its equal 4pq-^4pr-\-4qr, and bringing the term to 
the ollicr side as before. 

But since, from v;hat has been above laid down, we have 

v';>f4-\/<^'4-s/»-=a', and ^pqr=v h, 
if these be put for their equals in the last equation, it will become, by this 
substitution, 

,T 4 + 2fx2 —8h2x 4/2 -4g=0. 
Whence, comparing these coefficients with those of the givea equation, 
there will arise 

1fi=a;-^^h=b;f2^g=!:, or, 

And, consequently, by substifutirig these values in the assumed cubic equa- 
tion (2), we shall haye 

yZ^UyZ^ jg-(«2 _4c)jr=— (3). 

the three roots of which last eejuatidn, when substituted for /?, y, and r, io 
the formula j,= v';'4- v''" + v'9i will give, by taking each term of the ex- 
pression both in -J- and — , all the four values of x. 

Or, in order to render this result more commodious in practice, by freeing 
it from fractions, let ij=\z- Then by substitution and reduction, we sha'fl 
have the cerresponding equation 

^2 



174 



BiqUADlUTlC EqUATIOiNS. 



Find the three roots of the cubic equation ^^-fSa^^-j- 

{a'^ —4c)z=b" , by one of the former rules, before given 

for this purpose ; and let them be denoted by r', r' , and r". 

Then, we shall have 



When b is positive, 

2 

2 
'~ 2 ~ 



X- 



X- 



When h is negative, 



2 
— ^r-^s/r—s/r" 

2 

-\/'''-\/^"+v^^" 
2 

of the auxiliary 



jt 



jif 



Note. If the three roots r, r\ r 
cubic equation be all real and positive, the four roots of 
th» proposed equation will, also, be real ; and if one of 
these roots be ptvitive, and the other two imaginary, or 
both of them negative, and equal to each other, two of 
the roots of the given equation will be real, and two ima- 
ginary ; which are the only cases that produce real results. 

3. Given x« -25x2+ 60a; -36=0, to find the feur roots 
of the equation. 

Here a= — 25, 6=60, and c= - 36 ; 
Whence, by substituting thesevalues for their equals, in 
the cubic equation above given, we shall have 2^-2X25 
^2+ (252 -f 4x36)^—602, or ^^-SOz^-i- 7692=3600 : 



23 -f. 2«s2 +- (a2 — 4c)2=52 , (4) 
the three roots of which are each, evidentlj', four times those of ihe former. 
Hence using this instead of equation (3), and denoting its roots by r', r", r'", 
the last mentioned formula, taking each of its terms in -{- and — , as before, 
will give the values of a, as in the above expressions. 

JVotc. If we were to take all the possible changes of the signs, in this case, 
^hich the terms of the assumed fomiula admit of, it would appear that x 
should have eight different values; but it is to be observed, that, accoiding 
lO the first part of the above investigation, the product ^p X ^/ qY, >/''= 
t/Ti, or ^h\ and, consequently, that when b is positive, either all the three ra- 
dicals must be taken in^- , or two in— and one in-j- ; and when h is mega- 
dve, they muft either be all — ,-or two + and one — ; which considerations 
,-«du<e tie number tf roots to four. 



BIQUADRATIC EQ.UATIONS. 175 

the three roots of which last equation, as found by trial, 
or by one of the former rules, are 9, 16, and 25, re- 
spectively ; whence 

a:=i(-^9+v^l6-i-y25)=i(~3+4 + 5)=+3 

^=l(+v/9+yl6-y25)=i(+3+4-5) = + l 
And consequently the four roots of the proposed equa- 
tion are 1, 2, 3, and -6. 

EXAMPLES FOR PRACTICE. 

1. Given a;« — 55x2 — 302;-!- 504=0, to find the four 
roots, or values of x. Ans. 3, 7, —4, and —6. 

2. Given x*4-2x3— 7x2 _8a;=- 12, to find the four 
roots, or values of x. Ans. 1, 2, —3, and —2, 

3. Given x* —8x2 + 14x2 -{-4x=8, to find the four roots, 

or values of x, Ans. \ ^j,^' i~~ yc.» 

4. Given x* — 17x2 -20x— 6=0, to find the four roots, 

or values of x. Ans. \ _2'±./2' _2— /2* 

6. Given x*— 3x2— 4x — 3, to find the four roots, or 

values of x. Ans. \ fltt^ ^^' f ~t^ % 

6. Given x« — 19x3 + 132x2— 302x+200=0, to find the 

, ,, , ^ . < 1.02804, 4.00000 

iour roots, or the values oi x. Aos. < g 57653 7 '^9543 

7. Given x4—27x3+162xa+356x- 1200=0, to find 

., ,. , 1 r A ^ 2.05b08, -3.00009 

4he lour roots, or values of X. Ans. j .^ I'Sofi 14 "^086 

8. Given x* —12x2 + 12x — 3=0, to find the four roots, 
, . . i .606018, —3.907378 

•r values of x. Ans. J 2.858083, .443277 

9. Given x* — 36x2 +72x— 36=0, to find the four roots, 
, f. . i 0.87298^6, 12679494 

«T values ot x. Ans ^ 4 73205O6, -6 8729836' 

10. Given x« — 12x3+47x3 -72x+.36=0, to find the 
roots, or values of .t. Ans. 1^ 2, 3 and 6« 



176 RESOLUTION of EQUATIONS 

11. Given .T''-f24:cS — 114x3— 24a-4- 1—0, to find the 

1 e A ^+-1/197-14.2+^5 

roots, or values ot x. Ans. { ^,,r,n i • o /.- 

( —^I'dl—i-i, ^-—^/o 

12. Given a:* — 6a;3—58a-2-114x— 11=0, to find the 
roots, or the values of x. 

Ans. ±tv/3+f±v'(17±V-v/3:. 



OF THE 

RESOLUTION OF EQUATIONS 

BY APPROXIMATION. 

Eq,uations of the fifth power, and those of higher 
dimensions, cannot be resolved by any rule or algebraic 
formula that has yet been discovered ; except in some 
particular cases, where certain relations subsist between 
the coefficients of their several terms, or when the roots 
are rational ; and, for that reason, can be easily found by 
means of a few trials. 

In these cases, therefore, recourse must be had to some 
«f the usual methods of approximation ; among which 
that commonly employed is the following, which is univer- 
sally applicable to all kinds of numeral equations, what- 
ever may be the number of their dimensions, and though 
■ot strictly accurate, will give the value of the root sought 
to any required degree of exactness. 

* 

RULE. 

Find, by trials, a number nearly equal to the root 
sought, which call r ; and let z. be made to denote the dif- 
ference between this assumed root, and the true root x. 

Then instead of x, in the given equation, substitute its 
equal r ± z, and there will arise a new equation, involving 
oply 2 and knofrn quantities. 



b\ approximation. 17V 

Reject all the terms of this equation in which z is of 
two or more dimensions ; and the approximate value of 
z may then be determined by means of a simple equation. 

And if the value, thus found, be added to, or subtracted 
from that of *■ according as r was assumed too little or too 
great, it will give a near value of the root required. 

But as this approximation will seldom be sufficiently 
exact, the operation must be repeated, by substituting the 
number thus found for r, in the abridged equation ex- 
hibiting: the \'alue of z ; when a second correction of z 
will be obtained, which, being added to, or subtracted from 
r, will give a nearer value of the root than the former. 

And by again substituting this last number for r, in the 
above mentioned equation, and repeating the same process 
as often as may be thought necessary, a value of x may be 
found to any degree of accuracy required. 

JVote. The decimal part of the root, as found both by 
this and the next rule, will, in general, about double itself 
at each operation ; and therefore it would be useless as 
well as troublesome, to use a much greater number of 
figures than these in the several substitutions for the 
Talues of r.* 

EXAMPLES. 

1 . Given x^ -{■ x -f a; = 90, to find the value of x by 
approximation. 

Here the root, as found by a few trials, is nearly equal 

to 4. 



*It may here be observed, that if any of the roots of an equation fee 
irhole numbers, they maybe cletenained by substituting I, 2, 3, 4, <fec. suc- 
cessively, both in plvs and in mimis, for the unknown quantity, till a result i« 
obtained equal to that in the question ; when those that are found to succeed, 
will be the roots required. 

Or, since the last term of any equation is always equal to the continued 
product of all its roots, the num'b< r of these trials may be generally diminish- 
ed, by finding all the divisors of that term, ind then substituting them botli in 
plus and minus, as before, for the unknown quantity^ when those that give 
die proper result will be the rational roots sought ; but if notie of them are 
found to succeed, it may be concluded that the equation cannot be resolved 
by this mctliod ; the roots in that case, being either irrational »t imaginary. 



178 RESOLUTION of EQUATION^ 

Let therefore 4 = r, and r -^ z =x. 

Then x^ = r^-\-'2,rz-\-z^ =90. 

X =r -{-z 

And by rejecting the terras z^, 3rr2 and z^ , as small in 

comparison with z, we shall have 

r3 + r2 + r + 5r^z -{- 2rz +z,= 90 ; 

^, e0-r3 _r2 — r 90-64-16—4 6 

Whence z = ; ; — = ; =—=.10, 

3r2 -f-2r + 1 48+8+1 67 

And consequently x=4.1 , nearly. 

Again, if 4.1 be substituted in the place of r, in the last 

equation, we shall have 

90 — r3~.r2_r 90-68.921 — 16.81-4.1 

~ 3r2+2r+l 60.43+8.2+1 " ' 

And consequently a:= 4. 1 +.00283=4. 10283 for a second 
approximation. - 

And if the first four figures, 4.102, of this number be, 
again substituted for r, in the same equation, a still nearer 
value of the root will be obtained ; and so on, as far as may 
be thought necessary. 

2. Given a:2+20a;= 100, to find the value of a- bj^ ap- 
proximation. Ans. x=4. 1421356 

3. Given a;3 + 9x*+4a;=80, to find the value of a; by 
approximation. Ans. a:=2. 4721359 

4. Givena;*— 38T3+2I0a;2 + 538x+289=0, to find the 
value of X by approximation. 

Ans. .T=:30. 53565375 

5. Given a;= +6x4 — lOx^ - 1 12x2 _ 207a'+110 = 0, to 
find the value of x by approximation. 

Ans. 4.46410161 
The roots of equations, of all orders, can also be de- 
termined, to any degree of exactness, by means of the fol- 
lowing easy rule of double position ; which though it has 
Hot been generally employed for this purpose, will be 
found in some respects, superior to the former, as it can 
be applied, at once, to any unreduced equation, consisting 
of surds, or compound quantities, as readily as if it had 
been brought to its usual form. 



BY APPROXIMATION.- 179 



RULE. 



Find, by trial, two numbers as near the true root as pos- 
sible, and substitute them in the given equation instead of 
the unknown quantity, noting the results that are obtained 
from each. 
/. Then, as the dilTereace of these results is to the differ- 
ence of the two assumed numbers, so is the difference be- 
tween the true, result, given by the question, a>.d either of 
the former, to the correction of the number belonging to 
the result used ; which correction being added to that 
number when it is too little, or subtracted from it when it 
is too great, will give the root required nearly. 

And if the number thus determined, and the nearest of 
the two former, or any other that appears to be more ac- 
curate, be now taken as the assumed roots, and the opera- 
tion be repeated as before, a new value of the unknown 
quantity will be obtained still more correct than the first • 
and so on, proceeding in this manner, as far as may be judg- 
ed necessary.* 

* The above rule for Double Position, which is much more simple and 
eommodious tlian the one commonly employed for this purpose, is the same 
as that which was first given at p. 311 of the octavo edition of my Arithmetic 
-published in 1810. \ ' 

To this we may farther add, that when one of the roots of an equation has 
been found, either by this method or the former, the rest may be determined 
as follows : 

Bring all the terms to the left hand side of the equation, and divide the / 
whole expression, so formed, by the diftcrence between the unknown quanti- 
ty (a;) and th^ root first found; and the resulting equation will then be de- 
pressed a degree lower than the given one. 

Find a root of this new equation, by appioxi.mation, as in the first instance 
and the number so obtained will be a second root of the original equation. ' 

Tlien, by means of this root, and the unknown quanuty, depress the se- 
cond equation a degree lower, and thence find a third root; axid soon, till 
the equation is reduced to a quadratic ; when thi two roots of tliis, together 
WJth the former, will be the roots of the equation required. 

Thus in the equation .r3 — 15^2 +mx=^bO, die first root is found by a«- 
proximation to be 1.02804. Hence, ' ^ 

X— 1.02804(a3_l.';a;3 463a-50Gr2~13.9719Sa-4-4S.63627r=K). 

And the two roots of the quadratic equation, a;2 — 13.97196a=— 43 63627 
found in the usual way, are 6.57653 and 7. 39543. ' ' 

So that the three roots of the given cubic equation x^—\Bx2 +63a=5d 
are 1.02804, 6.57653, and 7. 39543 ; tlieir sum being=15, the coeflicier.t oi~ 
the second terwof the equation, as it ought to be when they are ri-^ht. 



J80 



RESOLUTION of EQUATIONS 



EXAMPLES. 

1. Given x<?+a;2 -f a;=100, to find an approximate value 
of X. 

Here it is soon found, by a few trials, that the value of 
X lies between 4 and 5. 

Hence, by taking these as the two assumed numbers, the 
operation will stand as follows : 

First Sup. Second Sup. 

4 . . X . • 5 
16 . . x2 . . 25 



Therefore 



64 

84 

155 

84 



Results 



125 

155 

100 

84 



71 : 1 :: 16 : .225 

And consequently a:=4+-226=4.225, Jiearly. 
Again, if 4.2 and 4.3 be taken as the two assumed num- 
bers, the oper.ition will stand thus : 

First Sup. Second Sup. 

17.64 . . x2 . . 18.49 



74.088 



Therefore 



95.928 
102.297 
95.928 



Results 
. . 4.3 
. . 4.2 



79.507 

102.297 
102.297 
100 






6.369 : .1 :: 2.297 : .03G. 

And consequently x=4. 3 --.036=4. 264, nearly. 
Again, let 4,264 and 4.265 be the two assumed num- 
bers ; then 

First Sup. Second Svp. 

4.264 . . X . . 4.26o 



I0.F81696 
77.526752 



X-' 



18.190225 
77.581310 



99.972448 Results 100.03653S 



BY APPROXIMATION. 



181 



Therefore 
100.030535 4.265 100 
99.972448 4.264 99.972448 



.064087 : .001 :: .027552 : .0004299 
And consequently 
.T=4.264-f-. 0004299 = 4. 2644299, very nearly. 
2. Given {^x^ — isy+x^ x=90, to find an approxi- 
mate value of x. 

Here, by a few trials, it will be soon iound, that the va- 
lue of X lies between 10 and 11 ; which let, therefore, 
be the two assumed numbers, agreeably to the directions 
given in the rule. 

Then 
First Sup. Second Sup. 

25 . . (ia-2_i5)2 . .84.64 
31.622 . . x^x . .36.482 



Hence 



56.622 
121.122 

66.622 



Results 
11 . 
10 . 



121.122 
121.122 
90 



31.122 : .482 
.482 = 10.518. 



64.5 : 1 : 

And consequently x=^\ 1- 
Again, let 10,5 and 10.6 be the two assumed numbera, 

Then 
First Sup. Second Sup. 

49.7026 . . (ix2-15)2 . . 55.830784 
34.0239 . . x^x . . 34.31 !099 



83.7264 

90.341883 . 
83.7264 



Results 
Hence 
10.6 
10.5 



. 90.341883 

. 90.341883 
. 90. 



6.615483 : .1 :: .341883 : .0051679 

And consequently 
x=10. 6— .0051o79=10. 5948321, very nearly. 

R 



t52 EXPONENTIAL EQUATIONS. 

EXAMPLES FOR PRACTICE. 

1. Given x^ + lOx^ -f 5.t=2600, to find a near approxi- 
mate value of x. Ans. =11.00673 

2. Given ^x* - I6x^-^40x^ —30x4-1=0, to find a near 
value of X. Aiis. x=l .2847!24 

3. Given i5^2x*-f 3r34-4x2 + 6.r=6^321, to find the 
value of X. Ans f.U4453 

4. Given 3/(7x3 + 4x2 )+.^(20x3 -10a)=28, to find the 
Talue of X. Ans. 4.510661 

b. Given ^(144x2-(x2 +20)2 ) + y(196x2-(a-2 +24)2) 
= 114, to find the» value of X. Ans. 7.123883 



Of exponential EQUATIONS. 

An exponential quantity is that which is to be raised to 
some unknown power, or which has a variable quantity for 
its index ; as 

X 1 

a^, a* , 3^ , or x^, &c- 
And an exponential equation is that which is formed be- 
tween any expression of this kind and some other quanti- 
ty, whose value is known ; as 

ax =b, x^='a. Sac. 
Where it is to be observed, that the first of these equa 
tions, when converted into logarithms, is the same as 

X log. a=b, orx=-; ; and the second equation x^=a 

log. a 

is the same as x log. x=log. a. 

In the latter of which cases, the value of the unknown 

quantity x may be determined, to any degree of exactness, 

by the method of double position, as follows : 

RULE. 

Find by trial, as in the rule before laid down, two num- 
bers as near the numbel- sought as possible, and substitute 
them in the given equation 

.1- log. x=log. «, 



EXPONENTIAL EQUATIONS. 183 

instead of the unknown quantity, noting the results obtain- 
ed from each. 

Then, as the difference of these results is to the differ- 
ence of the two assumed numbers, so is the difference 
between the true result, given in the question, and either 
of the former, to the correction of the number belonging 
to the result used ; which correction being added to that 
number, when it is too tittle, or subtracted from it, when 
it is too grent, will give the root required, nearly. 

And, if the number, thus determined, and the nearest 
of the two former, or any other that appears to be nearer, 
be taken as the assumed roots and the operation be re- 
peated as before, a new value of the unknown quantity 
will be obtained still more correct than the first ; and so 
on, proceeding in this manner, as far as may be thought 
necessary. 

EXAMPLES. 

1. Given a;i^=^100, to find an approximate value of a;. 
Here, by the above formula, we have 
X log. x=log. 100=2. 
And since x is readily found, by a few trials, to be nearly 
in the middle between 3 and 4, but rather nearer the Int- 
ter than the former, let 3.5 and 3.6 be taken for the two 
assumed numbers. 

Then log. 3.5=. 5440680, which, being multiplied by 
3.5, gives 1. 904238 =first result ; 

And log. 3.6= 5563025, which, being multiplied by 3.6, 
gives 2.002689 for the second result. 

Whence 
2.002689 . . 3.« . . 2 002C89 
1.904238 . . 3.5 . . 2. 



.098451 : .1 :: 002689 : .00273 
for the first correction ; vi'hich, taken from 3.6, leaves 
ar=3. 59727, nearly. 

And as this value is found, by trial, to be rather too small, 
let 3,59727 aad 3.59728 betaken as the two assumed num- 
bers. 



18 1 BINOMIAL THEOREM. 

Then log. 3.69728=0.555974243134677 to 1& places 
The log. 3.59727=0.555973036847267 to 15 places 

which logarithms multiplied by their respective numbers 
give the following products : 

1.999995025343512 } . *u * * iu i . c 
1.9999851226C2298 \ ^°^^ ^^"^ *° ^^^ ^^'^ ^g"'^" 

Therefore the errors are 4974656488 

and 14877337702 
and the difference of errors 9902681214 

Now since only 6 additional figures are to be obtained, 
we may oaiit the three last figures in these errors ; and 
state thus : as difference of errors 9902681 : difference of 
sup. 1 :: error 4974656 : the correction 502354, which 
united to 3.59728 gives us the true value of -x = 
3.59728502354*. 

2. Given a;*=2000, to find an approximate value of x. 

Ans. x=4.82783263 

3. Giveh (6a;)^'=96, to find the approximate value of x. 

Ans. a;=l b826432 

4. Given a;» = 123456789, to find the value of x. 

Ans. 8.6400268 
J. 

5. Given a;*— a;=(2.T— a:«)^, to find the value of x. 

Ans. X— 1.747933. 



OF THE 

BINOMIAL IHEOREM. 

The binomial theorem is a general algebraical expres- 
sion, or formula, by which any power, or root of a given 
quantity, consisting of two terms, is expanded into a series ; 
the form of which, as it was first propesed by Newton, 
being as follows : 



* The correct answer to this quoslion has been first g-iven by Doctor Adrain, 
in his edition of Hu lion's Mathematics, who plainly proves that Hutton's an- 
swer, which is the same as Bounvcastle"s, is iHCorrect ; Sec Hutt-on's J\fatke- 
maties, Vol. 1. p. 263. jY. Y. Edit. 



BINOMIAL THEOREM. 186 

,p+r«)„-=..-[i+-«+-(^)."+„(-^) 

,7>i — 2/1. . T 

(__),,, &c.] 

Or, 

w w Tjt m — n m — Sw 

m — 3n „ 

; DQ, &LC. 

Where p is the first term of the binomial, q the second 

Hi 

term divided by the first, — the index of the power, of 

?t 

root, and a, b, c, &.c. the terms immediately preceding 
those in which thev are first found, including their signs 
+ or-. " 

Which theorem may be readily applied to any particu- 
lar case, by substituting the numbeis, or letters, in the 
given example, for p, q,, m> and w, in either of the above 
fermulae, and then finding the result according to the rule.* 



* This celebrated theorem, which is of the most extensive US'* in algebra.. 
y and -'arious other branches of analysis, raaj be otherwise expressed as fol- 
lows : 

. x"^ '"ri.'«,^\ mm—n,x^ nim—nm—2nr 

(o4.3:)-=a-n4— -)4--.— (-)3 +-.— .—;: (-)3&C.j 

m 
Or, (a+T)-= 

m 'm. .T mtn-tn x inm+nm-^2n x 

m 
Or, (a^*x)'^= 

■m. m,a—x^ mm + na — *, mm-{-nn+2/7 a — x 

It may here also be observed, that if "' be made to represent any whole, ox 
Tractional number, whether positive or negative, the first of ihese expressionj 
may be exhibited in a more simple form 

wi(m — 1)(ot — ^2) [m — (n — l)]a"a:"»^ 

'■■■'■■■■ 1.2.3.4 7i ' 

"VVhere the last term is called the general term of the series, because if I. t.T 
i, 4, &c. be substituted successively for n, it will giv« al! the rest. 

k2 



186 BINOMIAL THEOREM. 



EXAMPLES. 



1. It is required to convert (o^+a;)^ into an infinite 
series. 

Here T=a^, q= — , — = -, or m = 1, and n = 2 .- 

a^ n 2 

whence 

m m ± 

in \ a X v 

- A^ = -X-X =- — =B, 

n 2 1 a3 2a ' 
m — n 1—2 X X x'^ 

in — 2n _ 1 — 4 x- x _ 3x^ _ 

3n~^^ 6~ ~2.4a3~ a^~2A.6aJ'~^^" 
vi-Sn _l-6 3x3 x_ 3.5a;< _ 

471 ~* ^ 8~ 2T476a5 ^ 2.4.6.8*7 ~^' 

TO— 4» __1-B 3.5x^ x_ 3.6.7^5 _ 

— ^-EQ — - 2.4.6.8"a'^a2"~2.4.6.8.lda9~^' 

&c. &c. &c. 

Therefore (a«4-a;)2 = 

.X x3 .3x3 3.5^4 3.5.7x« 

aH — 1- &r 

2a 2.4a3 2.4. 6a« 2.4.6.8 a' 2.4.6.8. lOa^ 

Where the law of formation of the several terms of the 

series is sufficiently evident. 

2. It is requiied to convert 7— XTTJ* *^^ *^^ equal (a+ 
i)"2, into an infinite series. 

Here p^^, q=-, and — = — 2, or ?»=— 2, and n=i ; 
"whence 

mm J 

p« = fa") n=a-2 = =A, 

m 2 16 26 

— AQ=— -X — X-= r=»> 

™ 1 flS a «3 



BINOMIAL THEOREM. 



IBI 



fn^n — 2 — 1 2b b 3b^ 

B€l= X rX-=— - = G, 

2n 2 a^ a a* 

1,1 — 2n —2-2 3ft» 6 46^ 

— CQ= — -— X— -X-= r=^' 

3» 3 a* a a^ 

m-3n —2—3 463 fc 56^ 

D^= : X --X-=— ^=E, 



An 
&c. 

Consequently 



4 a6 



a" a"^ 



&c. 



1 



1 26 362 ^46^ 66*_ 



«=» a-' 



a2 



, or its equal tf^ 



3. It is required to convert 

/a^ — x) 2, into an infinite series. 

Here 

P=a2 €i= , and — =-js-» or m= - 1 and n=2 ; 

whence 

711 m "1 I 

m_ll ^_^_ 

— AQ -X-X ~- -2^T-B, 

in-n -1—2,, X X Sx^ 

BQ= ; X— -X 



2n 
»ft— 2n 



2a 3 
— 1—4 3x2 



-cq: 



-X 



w 



X - 



2.4a« 



=c, 



3.5x3 



3n '" 6 ■ 2.4a* a2 2.4.6a'' 

i-.3» -1-6 3.5x3 a; 3.5.7x« _ 

■i>Q=— 17— Xttti^^X- 



^d, 



4n 
&c. 



8 



E, 



2.4.6a7" jj2 -2.4,6.809 
&c. -&C. 

Therefore 

(o2_x)^ a^2^a3>^2,4V^^2.4.6^a''>'^ 2.4.6.8^a»>' 

&.C. 



And 

3 ,x2v . 3.5 ,x3. 



«2 ^_ 1 X. 3 ,x2v 3.5 .x3. 3.5.7 .a^. 
: .v(o2-.x)'~" 2W'^2l4>a=^^~'"2.4.6^c* •'*"2.4.6 8 ^a^^ 

-fee. 



188 BINOMIAL THEOREM. 

4. It is required to convert ^9, or its equal (8+l.'3 
into an infinite series. 

Here p=-8, q;=-, and — =-, or m==l and n=3 ; 
8 . no 

^ ^ ^ Whence 

P"=(8)"=83=2=A. 

m _I 2 1 __ 1 _ 

m — n 1 — 3 11 1 



-B Q= X X- 



2n 6 3.22 023 3.6.2* 

m— 2n 1 — 6 1 1 



2w 9 3.6.2* 1^ 3.6.9.2V 

m—3n 1—9 6 1 6.8 

D Q3= X X — = ■ = £, 

An 12 3.6. g.S-' 23 3.6.9.12.21 • 

,n_4» 1 — 12 5.8 1 5.8.11 

.E^= _ X _——-—-- X 



bn 15 3.D.9.12.2i« 23 3.6.9.12.15.213 

&c. &;c. &c. 

Therefore ^9= 

11.5 5.8 . 5.8.11 



' 3.23 3.6.2" ' 3.6.9.27 3.6.9.12.21 <> ' 3.6.9.12 lo.'^i-' 

&c. 

5. ft is required to convert ^2, or its equal v/(^~rO» 

into an infinite series. 

* , , I 1,3 3.5 , 3.5.7 . 

Ans. 1^ — \ &c. 

2 2.4 2.4.6 2.4.6.8^2.4.6.8.10 

6. It is required to convert ^7, or its equal (8 — l)3i 
into an infinite series. 

Ape 2"- — -.— ^1^ 

' ' 3.22 3.6.2« 3.6.9.27 3.6.9. 12. 2* « 

7. It is required to convert ^240, or its equal 

(243 — 3)*, into an infinite series. 

1 4 4.9 4.9.14 

Ans 3— — — oiC 

6.33 5.10.37 510.15.311 5.10.15,20.315 

S. It is required to convert (a±a:)2 into an infinite series. 
> ~2a 2, 4a» ""2.4. 6a3 2.4.6.80* \ 



BINOMIAL THEOREM. 189 

0. It is required to convert {a±by into an infinite se- 
ries. 

¥(,^ ^ 262 2.563 2.6.86* J 

10. It is required to convert (a- 6)* into an infinite 
series. 

""■" I 4a 4.«a» '4.8. ISo^ 4.8.12.16a* > 

11. It is required to convert (a-f x)=' into an infinite 
series. 

'^ l^3a 9u3^92aa 92 120*^93. 12.15a^ ^ 

12. It is required to convert (1 — x)* into an infinite 
series. 

2x_2.3x3_ 2.3.8x3 2.3.8. 13x'' 

"^' 6'~6.10 "~ 5. 10.T5~ 5.10.16.20 

13. It is required to convert i^^^ '^^ equal 

(a±x)2 

(a±x) ^ into an infinite series. 

1 ^ , _x , 3x2 _ 3.5a;3 3.5.7x4 _ . > 

^°^- -1^-^2^+2:4^. -^2:4:6^"^iX6:8^ "*- ^'- 

14. It is required to convert ^, or its equal 

(a±x)3 

"i- . 
(a±x) 3 into an infinite series. 

at X 4x2 4.7x3 . 4.7.10x4 ™, ) 
J ^3aT3.6a2-^3.6.9a3^3.6.9.12a4 ^ 

15. It is required to convert j, or its equal 

(l+x)^ . 
"1- . 
/l+aj) ^ into an infinite series. 

X 6x3 6.9.c3 6.11.16x4 

^"'' ^"6+5lO~6lO.T5+6.10.15.20~^''' 



190 INDETERMINATE ANALYSIS. 

(^ Jim. Jl^ \.X. 
— - V, or its equeil 

(a+x) (a* — x^) ^, into an infinite series. 

, , X x^ x=» 3x* 3x5 6x« Sx'^ „ 

OF THE 

INDETERMINATE ANALYSIS. 

In the common rules of Algebra, such questions are 
iiaually proposed as require some certain or definite an- 
swer; in which case, it is necessary that there should be 
as many independent equations, expressing their conditions, 
as there are unknown quantities to be determined ; or 
otherwise the problem would not be limited. 

But in other branches of the science, questions fre- 
quently arise that involve a greater number of unknown 
quantities than there are equations to express them ; in 
which ir»taoces they are called indeterminate or unlimit- 
ed problems ; being such as usually admit of an indefinite 
number of solutions ; although, when the question is pro- 
posed in integers, and the answers are required only in 
whole positive numbers, they are. in some cases, confin- 
ed within certain limits, and in others, the problem may 
become impossible. 

PROBLEM 1. 

To find the integral values of the imknown quantities x 
and y in the equation 

ax — by=±c, or ax-\-by=!:c. 

Where a and b are supposed to be given whole nuna- 
bers, which admit of no common divisor, except when it 
is also a divisor of c. 



INDETERMINATE ANALYSIS. 191 



RULE. 

1. Let ■s.'h denote a whole, or integral number ; and re- 
duce the equation to the fornj 

by^c ■ c — by . 

x= — =-wn, or x= -wk. 

a a 

2. Throw all whole numbers out of that of these two 

exprcsisions, to which the question belongs, so that the 

numbers d and e in the remaining parts, may be each les6 

than a ; then 

dy±e , e — dy , 

— =Ts)h, or -=wh. 

a a 

3. Take such a multiple of one of these last formulae, 

corresponding with that above mentioned, as will make 

the coefficient of y nearly equal to a, and throw the whole 

numbers out of it as before. 

oy 
Or find the sum or difference of — , and the expression 

" ay 

above used, or any multiple of it that comes near — , and 

a 

the result, in either of these cases, will still be =wh, a 

whole number. 

4. Proceed in the same manner with this last result ; 
and so on, till the coefficient of y becomes = 1, and the 
remainder = some number r ; then 

^■^^^^- ^wh.=p, and y=.apz^r. 

Where p may be o, or any integral number whatever, 
that makes y positive ; and, as the value ofy is now known^ 
that of X may be found from the given equation, when the 
question is possible*. 

NoTR. Any indeterminate equation of the form 
ax — by='±.c, 



* This rule is founded on the roTious principle, that the sum, difference, 
or product of any two whole iiunioers, is a whole number; and that, if a 
number divides the whole of any other number *nd a part of it, it Trill slscr 
divide the remaining part. 



192 INDETERMINATE ANALYSIS. 

in which a and b are prime to each other, is always possi- 
ble, and will admit of an infinite number of answers in '■ 
whole numbers. 

But if the proposed equation be of the form 

ax-\-byz=:C, 
the number of answers will always be limited ; and, m 
some cases, the question is impossible ; both of which cir- 
cumstances may be readily discovered, from the mode of 
solution above given*. 



=■ 



EXAMPLES. 



1. Given \9x—\'ly=\l, to find x and y in whole num- 
liers. 

Here a:= — =^— r — =^wh., and also -~=wk. 

Whence, by subtraction, -~ — — ^— — =-^—-—z=znh. 

%— ^1 ,. 20?/— 44 ^ , V— 6 

Also, JL__x4=:-^-=,-2-f^=.A. 

And by rejecting y — 2, which is a whole number, 

19 ^ 
Whence we have y=l9p-\-6. 
. - 142/+11 14(19»+6)+ll 266p-{-95_ 
And ^=—^^ 19 ^9 



* That the coefficients a and b, when these two formulae are possihfe, 
should have no common divisor, which is not at the same time, a divisor of c, 
is evident ; for if assmfi, and 6=OTe, we shall have ax ■+■ by^mdx + mgyig= 

c ,• and consequently dx+ey=—. But d, e, x, y, being supposed to be whole 

(^ 
numbers —must also be a whole number, which it cannot be, except whan m 
7n 

is a divisor of c. 

Hence, if it were required to pay 100^ in guineas and moidoi'es only, the 
qnestion would be impossible ; since, in the equation 21 t -|. 2'7y=2000 which 
represents the conditions of the problem, the coefficients, 21 and 27, are each 
divisible by 3, whilst the absolute term 2000 is not divisible by it. See my 
Treatise of Algebra, for the method of resolving questions of this kind, by 
means of Continued Fractions. 



INDETERMINATE ANALYSIS. 193 

Up+5. 

Whence, if p be taken =0 we shall have x— 5 and y 
^6, for their least values ; the number of solutions be- 
ing obviously indefinite. 

2. Given 2x+3j/= 25, to determine a; and y in whole 
positive numbers. 

Hence, since x must be a whole number, it follows that 

— ^ must also be a whole number. 
2 

Let therefore —-^=wh==p; 

2 *■ 

Then 1 - ?/= 'Up, ot y=\— 2p, 
And since 

x=12-j/4-i^-=12-(l-2;7)+p=12+3/)-l, 

We shall have a;= 11 +3/7, and i/=l — 2p ; 
Where JO may be any whole number whatever, that will 
render the values of x and y in these two equations posi- 
tive. 

But it is evident, from the value of i/, that p must be 
either or negative ; and consequently, from that of x, 
that it must be 0, - 1, —2, or— 3. 

Whence, if ;3=0, p= - 1 , p= - 2, p= — 3, 

Ihen < . o r 17 

Which are all the answers in whole positive numbers 
that the question admits of. 

3. Given 3x=82/ — 16 to find the values of x and y in 
whole numbers. 

Here a:=-^ =2i/-5-| — ^^^^ — =wA ; or-^^ — =wh. 

3 33 

., 2v— 1 „ 4«-2 , y — 2 , 

Also-^ X2 = -^- — =y-\-±—-=n:h. 

3 3 ^ ' 3 

Or, by. rejecting y, which is a whole number, there will 

• 2/- 2 
remam — ^^—=^-h.=p. 



194 



hN'DETERMINATE ANALYbJi, 



.8;,. 



Therefore 7/= 3p4-2, 

3 3 3 

Where, if p be put = l, we shall have a-=8 and y—o, 
for their least values ; the number of answers being, as in 
the first question, indefinite. 

4. Given 21. T+17?/=2000, to find all the possible val- 
ues of X and y in whole numbers. 



Here 



2000-171/ 



Or, omitting the 95, 

2\y 



21 
Uy 



17?/ , 



Consequently, by addition, -— + 

21 



21 
5- 



\ly 4i/4-o 



21 



21 



= a-A. ; 



- M 4.v+5^^ 20?/+ 25 .4+20?/ 

Also, -^ X5=-^p- = l+-^— ==t4/i. ; 

4+ 20w 
Or, by rejecting the whole number 1, — —r—='wh. 

21v 4+20u w-4 

And, by subtraction,—- :> j . 



21 



.\nd .1 = 



21 21 

Whence ?/= 2 lp+4, 
2000 - 1 ly_ 2000 - 17(2 1/?+4) 

21 ^ 21 



-wh.^i 



92-17/j. 



Vvhere if p be put^^O, we shall have the least value ot 
2/=4, and the corresponding, or greatest value of a;=92. 

And the rest of the answers will be found by adding 21 
continually to the least value of y, and subtracting 17 from 
the greatest value of x ; which being done we shall obtain 
the six following results : 



a=92 
y=4 



75 
25 



68 
46 



41 

67 



24 
88 



7 
109 



These being all the solutions, in whole num,bers, that the 
question admits of, 

JVote. 1 . When there are three or mor.e unknown quan- 
tities, and only one equation by which they can be detei*- 
mined. as 

ax-\-hy-\-cz=d. 



INDETERMINATE ANALYSIS. 195 

it will be proper first to find the limit of the quantity that 
has the greatest coefficient, and then to ascertain the dif- 
ferent values of the rest, by separate substitutions of the 
several values of the former, from 1 up to that extent, as 
Q the following question. 

5. Given 3x+5(/-r72r=100, to tind all the different va- 
lues of .T, y, and z, in whole numbers'*. 

Here each of the least integer values of a- and y are 1: 
by the question ; whence- it follows, that 

-— 7 — 7—7 '^''' 

Consequently z cannot be' greater than 13, which is also 

the limit of the number of answers ; though they may be 

considerably less. 

By proceeding, therefore, as in the former rule, we shall 

have 

And, by rejecting 33—?/ — 22-, 
1-2?/— 2r , 3.V . \—2y—z_ y-\-\— z 

y-f-1 — z 
Whence^ — - — =p. 

And 2/=3/>-f-2^ — 1 ; 

And consequently, putting p=0, we shall have the least 

value of ?/=- — 1 ; where z may be any number, from I 

up to 13, that will answer the conditions of the question. 

When, therefore, ^=2 we have j = I, 



■) 



* If any indeterminate equation, of the kind above given, has one or more 
of its coefficients, as c, negative, the equation may be put under the form 

ax:-\-by=d-\.cz, 
in which case it is evident that an indefir.ile number of values may be given 
to the second side of tlie equation by means of the indefinite quantity z ; and 
consequently, also, to X and y in tlie first. And if the coetficients a, b, c, in 
any such equation, have a common divisor, while d has not, the question, as in 
the Urst case, becomes impossible. 



196 



INDETERMINATE ANALYSIS. 



Hence, bj taking z=2, 3, 4, 6, &c. the corresponding 
values of x and i/, together with those of ^, will be found 
to be as below. 



z= 2 


3 


4 


5 


6 


7 


8 


y= 1 


2 


3 


4 


5 


6 


7 


x^21 


23 


19 


15 


11 


7 





Which are all the integral values of ar, y, andr, that can 
be obtained from the given equation. 

Note 2. if there be three unknown quantities, and only 
two equations for determining them, as 

ax-\-by-^cz=d, and ex-\-fy-^gz=h, 
exterminate one of these quantities in the usual way, and 
find the values of the other two from the resulting equation, 
as before. 

Then, if the values, thus found, be separately substi- 
tuted, in either of the given equations, the corresponding 
values of the remaining quantities will likewise be deter- 
mined : thus, 

6. Let there be given x — 2y-\-z=b, and 2x'\-y~z=l, 
to find the values oi' x,y, and z. 

Here, by multiplying the first of these equations by 2, 
and subtracting the second from the product, we shall have 

3z - oj/=3, or z= — L^=l +2/+-^=w/i. ; 



2u .?v 
And consequently-^, or -^ 



=wh.=p. 



3 3 
Whence y==3p. 
And, by tiking p=l, 2, 3, 4, &c. we shall have y=3, 
6, 9, 12, 15, &c. and z=6, 11, le, 21, 26, &c. 
But from the first of the two given equations 
a:=54-2?,' — z ; 
whence, by substituting the above values for y and z, the 
results will give x=5, 6, 7, 8, 9, &c. 

And therefore the first six values of x, y, and z, are as 
below ; 



x=5 


6 


7 


8 


9 


10 


'2/=3 


6 


9 


12 


15 


18 


^ = 6 


11 


16 


21 


26 


31 



Where the law by which they can be continued is suffi- 
ciently obvious. 



INDETERMINATE ANALYSIS. 197 



EXAMPLES FOR PRACTICE. 

i. Given 3x=Sy~-l6, to find the least values of x and 
,f in whole numbers. Ans. a;=8, y=5 

2. Given 14x=5y-^l , to find the least values of x and 
y in whole numbers Ans. x=3, y=t 

3. Given 27a:=1600— 16?/, to find the least values of x 
and y in whole numbers. Ans. a;=48, ?/=19 

4. It is required to divide 100 into two such parts, that 
one of them may be divisible by 7, and the other by 1 1 . 

Ans. The only parts are 56 and 44 

5. Given 9.r4-13j/=2000, to find the greatest value of x 
and the least value of y in whole numbers. 

Ans. a:=215, y=^b 

6. Given lla:-f-5?/=^254, to find all the possible values 
of X and y in whole numbers. 

Ans. a-=19, 14, 9, 4 ; y=9, 20, 31, 42 

7. Given 17a-+ 19j/+ 2 1^=400, to find all the answers 
in whole numbers which the question admits of. 

Ans. 10 different answers 

8. Given bx-\-'iy-\^\ \z^=22i, to find all the possicie va- 
lues of X, y, and z, in whole positive numbers. 

Ans. The number of answers is 59 

9. It is required to find in how many different ways it is 
possible to pay 20/. in half-guineas and half-crowns, with- 
out using any other sort of coin ? 

Ans. 7 different ways 

10. I owe my friend a shilling, and have nothing about 
me but guineas, and he has nothing but louis-d'ors ; how 
must I contrive to acquit myself of the debt, the louis being 
valued at 17*. apiece, and the guineas at 21s. ? 

Ans. 1 must give him 13 guineasy«and he must 

give me 16 louis 

11. How many gallons of British spirits, at 12s., 155., 
and 18s. a gallon, must a rectifier of compounds take to 
make a mixture of 1000 gallons, that shall be worth 17s, a 
gallon ? 

Ans. nu,atl2j., nij at 155 ..and 777f at 13^ 



•s 



a 



198 INDETERMINATE ANALYSIS. 



PROBLEM II. 

To find such a whole number, as, being divided by other 
given numbers, shall leave given remainders. 

RULE. 

1 . Call the number that is to be determined x, the num- 
bers by which it is to be divided a, 6, c, &.c. and the given 
remainders/, g, h, &,c. 

2. Subtract each of the remainders from x, and divide 
the differences by a, b, c, &c. and there will arise 

X —f x-~g x — h 



a ' b ' 



-, &c. = whole numbers. 



x-f 

3. Put the first of these fractions — - =p, andsubsti- 

a ^ 

tute the value of a:, as found in terms of p, from this equa- 
tion, in the place of x in the second fraction. 

4. P'ind the least value of p in this second fraction, by 
the last problem, W'hich put=r, and substitute the value 
of X, as found in terms of r, in the place of x in the third 
fraction. 

Find, in like manner, the least value of r, in this third 
fraction, which put ^=s, and substitute the value of x, as 
found in terms of s, in the fourth fraction as before. 

Proceed in the same way with the next following frac- 
tion, and so on, to the last ; when the value of x, thus de- 
ermined, will give the whole number required. 

EXAMPLES. 

i. It is required to find the least whole number, which, 

oeing divided by 17, shall leave a remainder of 7, and 

ifhen divided by 26, shall leave a remainder of 13. 

Let X— the number required. 

x—l a: — 1 3 
Then and = whole numbers 

17 26 



INDETERMINATE ANALYSIS. 199 

re -7 ^ 

And, putting — — ==p, we shall have x=17/5-|-7. 

Wiiich value of x, being substituted in the second frac- 

17n+7— 13 ilp-G 
lion, gives =— |^=rfft. 



But it is obvious that— -|- is also =re>h. 

26 



26p 

, 2Gp 170-6 9/5 + 6 
And consequently^ ——=-j^=zvh . 

r. 9p+6 ,„ 27d-}-18 , p+18 , 

26 26 ^ ' 26 . 

. . , • P+18 

-Where, by rejecting /», there remains'— ——=ai/i. =r 

Therefore p=26r— 18 ; 
Whence, if r be taken = 1, we shall have p=8. 
And consequently a:=17p+7=I 7 X 8+7 = 143, the 
number sought. 

2. It is required to find the least whole number, which, 
being divided by 11, 19, and 29, shall leave the remain- 
ders 3, 5, and 10, respectively. 

Let x= the number required. 

Then——, ——and — — = whole numbers. 

x — Z 
I And, putting — — =p, we shall have a-=l lp+3. 

Which value of x, being substituted in the second frac- 

; llp-2 

ion, gives — —X — ='4'a. 

. lip — 2 ^ 22p-4 , 3»-4 

-Or — t- X2=— ^^^ =«+-£__= Tjyft, 

19 19 ^^ 19 

. 3p— 4 
And, by rejecting p, there v.'iU remain ■ -=r;/>, 

3t_4 18n— 24_]8p-5 
\lso by multiplication — — — Xd= — — -3 ■■ 5 



200 INDETERMINATE ANALYSIS. 

Or, by rejecting the 1 , — - =wh. 

But — ~- is likewise ='wh. 

Whence —^ E^==?Ljl=wh., which put =r, 

1 y 1 y 1 y 

Then we shall have 

p=19r— 5, and z=l 1 (I9r— 5) + 3=209r--52. 

And if this value be substituted for x in the third fraction. 

there will arise 

209r— 62 „ „ , 6?^— 4 , 

29 ^29 

Or, by neglecting 7r — 2, we shall have the remaining 

Qr — 4 
■part of the expression — 5^='^^*-/ 

But by multiplication, 

6r— 4 30r— 20 , r— 20 

29 29 ^29 ' 

r — 20 
Or, by rejecting r, there will remain =wh. which 

put =s. 

Then r =29s+20 ; where, by taking s=0, we shall have 

r=20. 

And consequently 

x= 209r-^62=209 X 20—52=4 1 28, 
the number required. 

3. To 6nd a number, which, being divided by 6, shall 
leave the remainder 2, and when divided by 13, shall leave 
the remainder 3. Ans. 68 

4. It is required to find a number, which being divided ' 
by 7, shall leave 5 for a remainder, and if divided by 9, 
the remainder shall be 2. Ans. 1 10 

5. 't is required to find the least whole aumber, which, 
being div ided by 39 , shall leave the remainder 1 6, aud when 
divided by 66, the remainder shall be 27. 

Ans. 1147 

6. It is required to find theleast whole numberrwhjch> 



DIOPHANTINE ANALYSIS. 20. 

being divided by 7, 8, and 9, respectively, shall leave the 
remainders 5, 7, and 8. Ans. 1727 

7. It is required to find the least whole number, which, 
being divided by each of the nine digits, 1, 2, 3, 4, 5, 6, 
7, 8, 9, shall leave no remainders. Ans. 2620 

8. A person receiving a box of oranges, observed, that, 
when he told them out by 2, 3, 4, 5, and 6 at a time, he 
had none remaining ; but when he told them out by 7 at 
a time, there remained 3 ; how many oranges were there 
in the box? Ans, 180 



OF THE 

DIOPHANTINE ANALYSIS. 

This branch of Algebra, vvhich is so called from its in- 
ventor, Diophantus, a Greek mathematician of Alexandria 
in Egypt. >vho flourished in or about the third century after 
Christ, relates chiefly to the finding of square and cube 
numbers, or to the rendering certain compound expressions 
free from surds ; the method of doing which is by making 
such substitutions for the unknown quantity, as will reduce 
the resulting equation to a simple one, and then finding the 
Talue of thnt quantity in terms of the rest. 

It is to be observed, however, that questions of this 
kind do not always admit of answers in rational numbers, 
and that, when they are resolvable in this way, no rule 
can be given that will apply in all the ca«es that may occur ; 
but as fir as respects a particular class of these problems, 
'relating to squares, they may generally be determined by 
means of some of the rules derived from the following 
formulas. 

PROBLEM I. 

To find such values of x as will make ^{ax^ ■\-hx-\'c) 
rational, or aa;'-+6x+ca=:a square. 



202 DIOPHANTINE ANALYSIS 



RCLE. 

i'. When the first term of the formula is wanting, or a 
=0, put he side of the square sought =?j ; then bx-\-c 
= n«. 

And, consequently, by transposing c, and dividing by 

n^ c 

the coefficient b, we shall have x= — - — j where n may 

be any number taken at pleasure. 

2. When the last term is wanting, or c=0, put the side 
of the square sough t=na-, or, for the sake of greater gene- 

rality,^ — ; then, in this case, we shall have ax--i-bx = 

m^x^ 






»2 

And, consequently, by multiplying by n^, and dividing 

by a-, there will arise an^x4-bn^ = m'x, and x= , 

where m and n, both in this and the following cases, may 
be any whole numbers whatever, that will give positive 
answers. 

3. When the coefficient a, of the first term, is a square 

number, put it =cZ2 , and assume the side of the square 

, . , ■ w , , , , , , , 2dm , m^ 

sought =dx-^ — : tlien, d2x'^'\~bx-\-c=d^x"-\ x-{- — . 

n n n^ 

And, consequently, by cancelling d^x^ , and multiplying 

by n2, we shall have bn^x-\-cn'^=2dmnx-\-in-, and x= 

bn^ — 2dmn 

4. When the last term c is a square number, put it = 

nix 
c2, and assume the side of the square sought = \-e; 

71/ 

then, aar2 -|-6a;+e3 = 1 x-^-e-. And, consequent 

n^ n 

ly, by cancelling c^ , and dividing by x, we shall have ax-{- 

, m^x , 2ein . bn^ —2emn 
0= — --1 , and a;= —. 



DIOPHANTINE ANALYSIS. 2Q3 

5. When the given formula, or general expression, 

can be divided into two factors of the form fx-{-g and/ix-j- 
k, which it always can when 6^ -^400 is a square, let there 

he t^ken {fx+g)X{hx-}-k)=—{fx+gy ; then, by re- 

duction, we shall have x=-, : where it may be 

observed, that if the square rootof 62__4ac, whenrationalj 
be put =J', the two factors above mentioned, will be 

„.+'^, and .+'±i 

"And, consequently by substituting them in the place of the 
former, we shall have 

am^ (6 -J')~ «2 (b-\-^) 

2a(re2— am2) 

6. When the formula, la=t mentioned, can be separated 
into two parts, one of which is a square, and the other the 
product of two factors, its solution may be obtained by 
putting the sum of the square and the product so, formed, 

equal to the square of the sum of its roots, and — times 

n 

one of the factors, and then finding the values of x as in 
the former instances. 

7. These being all the cases of the general formula that 
are resolvable by any direct rule, it only remains to ob- 
serve., that, either in these, or other instances ol a differ- 
ent kind, if v/e can find,, by trials, any one simple value of 
the unknown quantity which satisfies the condition of the 
question, an expression may be derived from this that will 
furnish as many other values of it as we please. 

Thus, let p, in the given formula ax- -\-bx-\-c, be a value 
of X so found, and make ap'-\-bp-\-c=q^ . 

Then, by putting z=ij-\-p, we shall have ax^-j-hx-{-c= 
°{y-^py ■i-Ky-{-p)-\-c=ar +{"2ap+b)y-{-ap^ +bp-\-c, or 
ax~-{-bx-\-c=ay^ -\-{2ap-\-b'y-{-q2 . 

From which latter expression the values of y, and con- 
sequently those of X may bo found, as in Case 4. 



204 DIOPHANTINE ANALYSIS. 

Or, because c=»q^ — bp — a/)^, if this value be substitut- 
ed for c in the original formula a-x'^+bx+c, it will become 
a{x'^—p^)+b{x— p)-\-q^ , or 

92 4- (a; — p) X{ax + ap-\-h)—a square ; 
which last expression can be resolved by Case 6. 

Itf may here, also, be farther observed, that by putting 

1/2 , , . z—b 

the given formula ax"+bx+c= ^, and takmg ^=^-^ \ 

we shall have, by substituting this value for x in the former 
of these expressions, and then multiplying by 4a, and trans- 
posing the terms 0^2 + (6--4ac)=z2 ^ or,putting,for the sake 
of greater simplicity, 62 — \ac—h\ this last expression may 
then be exhibited under the form ai/2+6'=z2^ ^vhere it is 
obvious, that if a'^j^^\^{h^~^ac), or its equal ai/2+fc', can 
be made a square, ax2-f 6a;+c, will also be a square. 

And as the proposed formula can always be reduced to 
one of this kind, which consists only of two terms, the pos- 
sibility or impossibility of resolving the question, in this 
state of it, can be more easily perceived. 



EXAMPLES. 

1. It is required to find a number, such that if it be mul- 
tiplied by 5, and then added to 19, the result shall be a 
square. 

Let .T= the required number ; then, as in Case 1, hx-\- 

19=n,2, or x= ; where it is evident that n may be 

5 

any number whatever greater than y/19. 

Whence, if n be taken =5, 6, 7, respectively, we shall 

25 — 19 36-19 ^, 49-19 „ ^, 

have X— — - — —\\, or — -— =3|, or — - — =6 ; the 

latter of which is the least value of x, in whole numbers, 
that will answer the conditions of the question ; and con- 
sequently 5a; -f 19=^5X6-1-19=30+19=49, a square 
number as was required. 

2. It is required to find an integral number, such that it 
shall be both a triangular number and a square. 



DIOPHANTINE ANALYSIS. 205 

it IS here to be observed, that all triangular numbers are 
of the form — ; and therefore the question is reduced 

to the making — - — , or its equal — — a square. 

Where, since the divisor 4 is a square number, it is the 
same as if it were required to make 2a;2-|-2a; a square. 

Let therefore 2x'^-{-2x=(^ — ) = — ^^, agreeably to the 

method laid down in Case 2. 

Then, by dividing by x-, and multiplying the result by 

n^ , the equation will become 2n~ x-^-^'in'-^ ==ifi,^ x, or (m^ — 

2n- 
2n-) x^=2n" ; and consequently x— — - — -— - ; where, if 

^ J" — 2?t- 

X^ — r-jC 

nbe taken =2, and m=3, we shall have a;=8, and — - — 

64-4-8 72 . . 

= = — =36, which is i*ne least integral ti ian,'»;ular 

2 2 a o 

wumber that is at the same time a square. 

3. It is required to find the least integral number, such 
that if 4 tifnes its square be added to 29, the result shall 
be a square. 

Here it is evident, that this is the same as to make 4x" 
+29 a square. 

And, as the first term in the expression is a square, let 

4x2+29= (2x-| — ) =4x2+ — 3--f- — ; agreeably to Case 

3, 

„, 4m , in^ „„ 4m. ^^ ni* . ' 

Then, — x+— =29, or — x=:29 ; and conse- 

n n^ - n n- 

quently x= ; where, if m a.'.d n be each taken 

^ -' 4mn 

29— I 
= 1, we shall have x= =7, and 4x2+29=4X49+ 

29=225=(16)' , which is a square number, as was re- 
quired. 

4. It is required to find such a value of x as will ma^e 
5x+l a square. 



:c 



206 BIOPHANTIxVE ANALYSIS. 

Here the last terra 1 being a square, let these be taken, 
according to Case 4. 

Then, by rejecting the 1 on each side of the equation, 

and dividing by X, we shall have 7x — 6= — x , and 

° "^ n^ n 

consequently a'= — - — —-5-; where, it m and ?i be each 

2—5 3 

taken =1, the result will give x= =-=1^, or by tak- 

^ 1—7 6 - 
43 45 

xng n=iS, and jn=8, v/e shall have x^-— — -^r=3, which 

ol — 63 

makes 7X32-3X3+1=49^72, as required. 

6. It is required to find such a value of x as will make 
8x^-\-\4x-\-6 a square. 

Here, by comparing this expression with the general 
formula aa-2 -|-^x-[-<;, we shall have a=8, 6=14, and c=6. 

And as neither a nor c, in the present instance, are 
squares, but b^ ~4ac=196— 192^=4 is a square, the given 
expression can be resolved, by Case 6, into the two follow- 
ing factors 8.r-}-6, and x-\-\. 

Let, therefore, 8a:2 + 14x+6=(8x+6)(a;+l)= — j(.r+ 

1)2, agreeably to the rule there laid down. 
Then there will arise, by dividing each side by x-f:K 

8a;-}-6=— (x+1). 

And, consequently, by multiplication and reduction, we 

shall have, in this case., x=- ; where it appears. 

0?l2— Wl2 

that, in order to obtain a rational answer, -— must be less 

n2 

than 8, and greater than 6. 

Whence, bv taking m~5, and 7i = 2, we shall have x = 
25-24 1 ' . ^ , 8 , 14 , - 400 .20.2 
32:12-5=7' ^'^'^^ "'^^^^ 4U+T + ^ = 19-=Ct) ^^''' 
cjuired. 



DIOPHANTINE ANALYSIS. 2©7 

0. It is required to find such a value of x as "tviil make 
il.x'^ —2a square. 

Here, by comparing this with the general formula oj:- . 
■rbx-{-c, as before, we shall have a = 2, 6 = 0, and c= — 2, 

And as neither o nor c are squares, hutb^ — 'iac= — 4ac 
— -.4(2 X —2) = 16 is a square, the root of which is 4, 
the given expression can be resolved, by Case 5, into the 
two factors 2x — 2, and a-f ' . or 2(x-— 1), and (.r+l), 
which is evident indeed, in this case, from inspection. 

Let, therefore, 2x--2=2(a;-l)X(x+l)=— (xf 1)S 

fV 

agreeably to the rule ; and there will arise by division 
2x — 2= — (a;+l). And, consequently, by multiplication, 

aad reducing the resultjWe shall havex=7r— ; where, 

by taking n= 1, and in=i, we shall have x=3, and 2x^ — 
2=18 — 2= 16 = (4)2, or taking 72 = 2, and m=3, the re- 
sult will givt a-=-- 17. 

But as X enters the problem only io its second, +17 
may be taken instead of — 17 ; since either of them give 
2x2 -2= 676 = (24)2. 

7. It is required to find such a value of x as will raal^e 
5r2-l-3(3a;-f-7 a square. 

Here, by comparing the expression with the general 
formula, we shall have a — 5, 6=36, and c=7. 

And as neither a nor c are squares, but b- —4ac=l296 

— 140 = J 156=(34)2 , is a square, it can be resolved, as in 

the last example, into the two factors 5x-rl( and a--|-7. 

m- 
Whence, putting 5x2 +3Gx + 7 = (.':r-fl)X(a:+7)== — 

(x-^-iy, there will arise, by dividing by x-f 7, 5x4-1=^ 

— (.r+7). 

And, consequently, by multiplication, and reducing the 

, „ , 7m2-?z2 
resuUmg expression, we shall have x=-- : waere 



208 DIOPHANTINE ANALYSIS. 

taking m=2, and n=l, the substitution will give ae= 

12^-1=.,27, which makes 5X(27)2+36X27-!-7=4624 
SXl-4 ^ ^ 

«=(GC)", as required. 

8. It is required to find such a value of x as will make 
6x2-l-13a-4-lu a square. 

Here, by comparing the given expression with the ge- 
neral formula ax^ -{-bx-^-c, we have a=6, 6=12, and c^= 
10. And as neither «, c, nor 6^ — 4ac, are squares, the 
question, if possible, can only be resolved by the method 
pointed out in Case 6. 

In order, therefore, to try it in this way, let the firsj; 
simple square 4, be subtracted from it, and there will re- 
DDain, in that case, Qx^ -f 13x4-6. 

Then, since (13)2 -4(6 x 6)=169- 144=25, is now a 
square, this part of the formula can be resolved by Case h, 
into the two factors ; 

3x4--2, and 2x+3. 

Whence, by assuming, according to the rule, Bx^-\-\Sx 

+ 10=44- (3x+2)X(2x+3)= ^ 24- -(3x-f 2) T =4 4" 



— C 3x4-2)-!-— ('3x+2y, we shall have, by cancelhng 

4m 
the 4, on each side, and dividing by 3x-f-2 ; 2x-t-3 — — 



n 



m 



2 



And, consequently, by multiplying by n^ , and transpos- 
ing the terms, we have 2n^x — 3m'^x=imn-\-27n'' — Sn^, or 

^ 2n2— 3m'3 

Where putting m==^-2, and n=3, the result will give x= 

244-8 —"7 5 

-~^ —=^-, or if m be taken = 13, and «=17, we shall 

18-12 6 

4X17X134-2X(13)3-3X<17)2 355 ^ 
2(17)2-3(13)2 71 



DIOPHANTINE ANALYSIS, 209 

Which makes 6X(6)2 + 13X 6 +10=225=(15)S as 
required. 

9. It is required to find such a value of x as will make 
13a-2 + ]5a:+7 a square. 

Here, by comparing this with the general formula, as 
before, we have o=13, 6=15, and e=^l. And as neither 
a, 6, nor b^—Aac, arc squares, the answer to the question 
if it be resolvable, can only be obtained by Case 6. In 
order, therefore, to try it, in that way, let (1 — .t)^ or 1 — 
2x4-.r- be subtracted, from the given expression, and 
there will remam li2;r2 + 17.T+6. 

And as (17)2—4(6X12), which is = 1, is now a square, 
this Mrt of the formuXi can be resolved by Case 5, into 
•the two factors 4a.-l-3 and 3a; + 2. Whence, assuming 13 

22 + 15x+7=(I-a:)24-(4.r+3) X {^x-^'i)=\{l-x) + 

'^(3a:4-2)J'=(l-:r)^+^"(I-^)X(3:r+2)+^^(3a: + - 

2)2, we shall have, by cancelling (1 — xY , and dividing by 

Sa;+2 : 4x+3=— (1 -a;)-| — -(3a:+2); and consequent^ 

ly, by multiplying by n- , and transposing the terms, there 
will arise 4n2x+2m«a;— 3//4-x= 2m«+2m2 — 3-^'>', or .t = 
2mn4-2m2 -3re2 
4uH-^imr^3m2 ' 

Where puttitig ?« and n each=^l, we shall have x = 
2+2-3 1 , . . , ^3 , 15 13 45 63 121 

~( "^ ) ' 33^ requireo. 

10. It is required to End such a -value of x as will make 
7x2+2 a square. 

Here it is easy to perceive that Eeither of the former- 
rules will apply. 

But as the expression evidently becomes a square v/her. 

-=1, let, thcrefore,,j:=l+y, according io case 7, and wt 

■^hall have 

7a;2+2=9+l4i/+7y2, .. 

t2 



210 ' DIOPHANTINE ANALYSIS. 



Or, putting 94-14?/+7?/2==(^3-|--j/)', according to the 
rule, and squaring the right hand side, 9-\-l4y-^ly^=9-{- 

Hence rejecting the 9's and dividing the remaining terqtis 

hyy,weh-Ave'ln~y-\-l4n^==6mn-\-m'-y; and, consequent- ^ 

6/rm — 1471^ , , Qmn—liti^ , 
lv»'V=- — — , and a-=l-i ; where it is 

evident that in and n may be any positive or negative num- 
bers whatever. 

I.^, for instance, 7n and n be each taken = 1, we shall 

4 1. ♦ 

have y= — and 2:=—-. Or, since the second power of 

X only enters the formuhT, we may take, as in a former in- 
stance, .r=-i-, which valae mtikes 7a=-i-2=§+2-=^J+ V' 
=2-5 a square. 

Or, if m=3 and «=— 1, Ave shall have a;=17, and 7x- 
4-2=7 X( 17)2 +^i=2026=(45)2, a square as iai-fore. 

And by proceeding in this manner, we may obtain as 
many otlier values of x as we please. 



^^ PROBLEM II. 

To find such values of x as will make y/(ax'^-\-bx^-\-c 
x-\-d) rational, or ax^-{-bx--\-cx-\-d= a square. This 
problem is much more limited and difficult to be resolved, 
than the former ; as there are but a few cases of it that 
-admits of answers in rational numbers ; and in these the 
rules for obtaining them are of a very confined nature ; 
being mostly .such as are subject to certain limitations, or 
that admit only of a few simple answers, which in the 
instances here mentioned, may he found as follows. 

RVLE. 

1« When the third and fourth terms of the formiila are 
wantiiig, or c and d are each =0, put the side of the sguar^ 
sought ==nx, then aa'3 4-ia:2=7»2j:3. 



DIOPHANTINE ANALYSIS. 211 

And, consequently, by dividing each side of the equa- 

, , n^ — b , 

tion bv a;2, we shall have ax+6=7i-, orx= , where 

■' a 

n may be any integral or fractional number whatever. 
2. When the last term d is a square, put it =e, and as- 

C J 1 

sume the side of the required square =e-\-—-x ; and the 

reversed formula is e^ -{-cx-i-bx^ -\-ax^=ze^ -Jrcx-\----x'' , 

Whence, by expunging the terms e^-}-cx, which are com- 
mon, and dividing by a;2,we shall have, 4ac'^x-{-4be~ = c'^ ; 

c2..4be2 
and, consequently, x= — - ^ . 

c , 46e2_c2 
- Or, if, in the same case, there be put e-\-—x-\ r— — a-^ 

for the side of the required square, we shall have, by squar- 

ang, e2-|-cj:+6x2-f aa;3=e2-j-ca;+6x2H — ^ — — -x^ -r 

i —x^ . And as the first three terms (e- -{-cx-^bx^ ^ 

64e6 ^ 

are now common, there will arise, by expunging them, ana 
then multiplying by 6\e^, G-iae^x^^^Sce^^-ibe" — c'^)x^-{- 
(46e2 — c2)3a;*. 

Whence, by dividing each side of this last equation by 
x', and reducing the result, we shall have 
_64(»c6_8ce3(46e2_c2) 
^ — (ibe^-c^)^ ' 

which last method gives a new value of x, different from 
that before obtained. 

3. When neither of the above rules can be applied to 
the question, the formula can be resolved, by first finding, 
by trial, as in the former problem, some value of the un- 
known quantity that makes the given expression a square ; 
in which case, other values of it may be determiricd froiri 
this, when they are possible, as follows : 

Thus, let p be a value of x so found, and make 



il^ DIOPHANTINE ANALYSIS. 

Then, by putting x—y-\-p, we shall have ap^-r-bp^+cp 
'^d:=a(y-hpy+y,y-{-py-{-c{y+p)-hd=ay^-{-{3ap+b)y^ 
+i'ip^+'^p-{-c)yi-up^-\-bp^-\-cp-\-d, or ax3-\-bx^-i-cx-{- 
d=ay^+{3ap-\-b)y-^+{^p^-\-2p+c)y-{-gK 

From which latter form, the value of y, and consequent- 
ly that of X, may be found by either of the methods given 
in Case 2. 

It may also be further remarked, that, if the given for- 
mula, in any case of this kind, can be resolved into factors, 
such that one of them shall be a square, it will be sufficient 
to make the remaining factor a square, in order to render 
the whole expression so ; since a square, multiplied or di- 
vided by a square, is still a square. 



EXAMPLES. 

1. It is required to find such a value of x as ■will tnalce 
ilx^-\-Bx^ a square. 

Let the given expression I lx^-\-3x^ ~n^x- : agreeably 
to Case 1. 

Then, by dividing by a-^ , we shall have lla-fS^n^; 

n- —3 
and, conseqtienlly-, a;= ; where n may be any num- 

ber, positive or negative, that is greater than V-S- 

Taking, therefore, n^=2, 3, 4, 5, &c. respectively, we 

shall have, in this case, -=^— , --, — , or 2, the last of 

which is the least integral answer that the question admits 
of. 

2. It is required to find s'ach values of x as will make 
x^ ~2x^ -\-2x-\-l a square. 

Here, the last term 1 , being a square, let 1 -fSx ^ Sz^ -^ 
x^=[l-\-xy = 'i-\'2x-{-x" , agreeably to the first part of 
Case 2. 

Then, since the first two terms, on botb sides of the 
equation, destroy each other, we shall have x^ — 2x-^=x- , 
ora:3=3x2, and conseq'iently x=3 ; which, by snbstitu* 
lion, makes i+2j;— Sa;^ J-x3=l-}-6-18-f27=I6, asguare 
•as required. 



DIOPHANTINE ANALYSIS. 2i5 

Again, by putting a:=?/4-3, according to Case 3, we shall 
have 1 + 2a;- 2x2 -^x^=l+2{y-{-3) - 2(2/+3)^ + (i/-(-3)3 

And, c6nsequent]y, by making \6-\-l'7y-r7y^-{-y^ = \^-i: 

17 • 289 

+— -t/)2 = 1 6 -{- 1 7y-\—- — y- , agreeably to the first part of 

Case 2, by cancelling l6-\-lly, there will arise '7y^'Ty^ = 

289 . „ 280 
— — ■j/2. orw+7= . 

289 ^ £89-446 159 

Whence v= 7= = > '^^^^ x = 3-- 

-^ 64 t)4 6i 

159 192-159 33 . . , . 

' = = — , for another value ol x. 

64 64 64 

"Which number, being substituted in the original formala, 

, . . ^ « o . , 429026 .655.2 
make l4-2x-2«;9+3:3 = ^^— =(— -) , a square, as 

before. 

5, It is required to find such values of x as will make 
3a;3 — 5a;2-f 6a;+4 a square. 

Here, 4 being a square, let 4-{-6x — 5x^-\-3x^={2-{' 
|x)'<=4 + 6a:4-f-^^5 as in the first part of Case 2. 

Tiien, since the two first terms on each side of the 
equation destroy each other, we shall have 3x^ —5x^=^- 
z^ , or 3a; — 5=f ; and, consequently, in this case x'=-- 
5+f^29 

3 12* 

Whence (2+f X^t;) =^24-^)' (-r) a square, as was 
12 o'G 

required. 

Or, by the second method of the same Case, let A-\-Qx 

2Q 87 841 

— 5.t2 +3x3 = (2_j_| X la;2 )2 =4+6x— 5a;2 - x^-^~- 

1 o 1 -^Ou 

;:^ ; then, as the three first terms on each side of this 

equation destroy each other, we shall have — -x* x^ 

^ J ' 256 16 

841 87 
—Sps, or-— -X- ——3. or 84]x— 1392=768 : and con- 
25o lij 



21-4 DIOPHANTINE ANALYSIS. 

,, 1392-f768 2160 ,. , . 

sequently, x = -—- = , winch is anolher value 

^ •" 841 84 i 

of X, that, being substituted in the original formula, will 

naake it a square. 

4. It is required to find such values of % as will make x-' 
+ 3 a square. 

Here, it is evident, that the expression is a square when 

.'r=I. Let therefore x=l-^y, and we shall have 3-\-x- 
=4+32/4-31/2+^3. 

And as the t'lrat part of this is a square, naake accordine 
to the first part of Case 2, 4+3!/+32/2+i,3 = (2 + fw)- =4 
+3?/+/^2/2. Then, because the first two terms on each 
side of the equation destroy each other, we shall have v^ 

^\ hence j/=-^3=-^-=---, aad x=l~ ^-^ 

16 — 39 23 

— — : — =— -r- ; which is a second value of x. 

3 39 
Again, let 4+5y+3y^-i-y^=(2-}--y+—y^y=4-}-3y 

117 1521 

+31/2+2^34.^^2^4^ according to the second part of 

Case 2. 

Then as the first three terms on each side of the equa- 

1521 117 
4ion destroy each other, we shall have y*-\ -1/^=?/", 

^ ' 409tj 128^ -^ 

1521 , 117 , 

or — -,— vH =1. 

4096^ ^128 

352 352 1873 

Whence, also, y=- — —, and a;=I-j ^= , which 

•^ 1521' ^ 1521 1521 

is a third value of x. 

And by proceeding in the same way with either of these 
new values of x as with the first, other values of it may be 
obtained ; but the resulting fraction will become continu- 
ally more complicated in each operation. 



BfOPHANTfiNE ANALYSIS, 2lo 



PROBLEM III. 

To find such values of .r as will make ^(aa;*+6;c3 4- 
€x--{-dx-\-e) rational, or ax* -{-bx^-^-cx^ -^ dx-\-e=-- asqnare. 

The resolution of expressions of this kind, in which the 
indeterminate, or unknown quantity, rises to the fourth 
power, is the utmost limit of the researches that have hi- 
therto been made on formula affected by the sign of the 
square root ; and in this Problem, as well as in that last 
given, there are only a few particular cases that admit of 
answers in rational numbers ; the rest being either impos- 
sible, or such as afford one or two simple solutions ; which 
may generally be found as follows : 

RULE. 

1 . When the last term e, of the given formula, is a square, 
put it •— /3, and make /^ + rfa; + cx^ -\- bx^-\-ax'^={f-{- 

64/« 
Then, by expunging the first three terms, which are 
common to each side of the equation, there will remain 

bx^+ax^ = -^ ^ ^r^-fi — - — <- x^, And conse- 

quently , by dividing by x' , and reducing the result, we shall 
have ,.- 6-^y^-8cy-'(4cA^-^--) 

2. When the coefficient, a, of the first term of the for- 
mula, is a square, put it =g-i and make g^x*-{-hx^-\-cx^ 

h 4crr2_52 

A-dx-{-e={gx^+—x-\—^^ — Y=gix^-\-hx^ 4- cx2 + 

fc(4cg2 -h^y_ , , (4cg2 -&2)2 
8^* 64^6 

Then. dx-\-e~^—^- ^z-f^— 2 -J- ■ and con- 

8o-* 64^*= -' 



216 DIOPHANTINE ANALYSIS, 

(4cs^-b"y-64cg^ ,. , ^ 

seqviently, oc=-~-^ — - , „ . . — „ , - . ; which form like 

wise fails under similar circumstances with the former. 

3. When the first and last terms of the formula are both 
squares, put a=g^, and e—f^, and make/^+^^+c^'-f- 

^x^+g^xK Then cx^-\-bx^ = (2/^+ ■^)x^+^x . 

And, consequently, x=^ — /{kf — jf^ — • 

Or, because g enters the given formula only in its second 
power, it may be taken either negatively or positively ; 

and, consequently /We shall have x=- — % • >, ^ , , . — -• So 

^ -^ . . f{¥+dg) 

that this mode of solution furnishes two different answers. 

Also, if there be taken for another supposition/^-f-cZx- 
+c.t2+6:c3 Jrg^-x^^{f+^x-\-gx^^y=.f2j^MxJr{2fg-^ 



% 



fc2 bf 

— -)x'-^hx^-\-g^x*, hence by cancelling, dx'\-cx^=^-^ x 

+(2/^4— ^)a;2 ; and consequently, a;=— ^^^— -i^— " 

And because/ enters the given formula only in the se- 
cond power, it may be taken either negatively or positively; 

g(h^-{bf) 
and, consequently, we shall also have x=^-—- — ^-r-;^^^ r. 

. . . . f'^-.^n^^/^-^-o 

So that this solution likewise furnishes two values of x, 
which are each different from the former. 

But these forms all fail under similar circumstances with 
those of the second Case. 

4. When neither the first nor the last terms are squares, 
the formula cannot be resolved in any other way, than by 
first endeavouring to discover by trials, some simple value 
of the unknown quantity that will answer the conditions 
of the question ; and then finding other values of it, accord- 
ing to the methods pbinted out in the two last probleaw. 






DIOFHANTlN£l ANALYSIS. 217 

Thus, let p be a value of x so found, and make 0^*4* 

Then by putting x=y-^p, we shall have ap*-j-^/>3-j-cp2 

+ dp-\-e=a{y+py+b{y+t*y-{-c{y+pY + d{y+p)+e= 
ay'^ + {ap-i-b)y^ + {6ap^+3bp +c)y^+{4ap3-\- 3ijp2^0cp 
-\-d)y-\-ap*4-bp^-{-cp^+dp+e, or ax*+bx^+cx^-\rdx4' 
€~ay^ + lap-{-b)y^-\-{6ap^+3bp-\-c)y^ + {4ap^ + 3bp^+'2 
cp-{-d)y-\-q- . From which last formula the value of «/, and 
consequently that of x, may be found by Case 1. 

EXAMPLES. 

J. It is required to tind such a value of x, as will make 

1 — 2x-{-3x-—4x^ + ox* a square. 

Here, the first term 1, being a square, let l — '2x-\-3:^^ 
~4x-3 + 5a;*=(l-.'r+.'c2)23=i-2a;+3x2_2i3+a:*, agree- 
ably to the method in Case 1 . 

Then we shall have ox'^ — 4.T^=a'* ~.2x' . 

And, consequently, .^.-c — 4=.^• — 2 ; whence a:=|=A. 

And consequently, 1— 2a-+3a;2-4x3-{-5x« = l -l+a^i 
59" 
-! — = — : which is a square number, as was required. 
~1G 16' ^ '1 

2. It is required to find such a value of a-, as will make 
4x* —2x^—x^-^3x — t a square. 

Here the first term being a square, let 4x* — Sa:^ — x^-f- 

5 5 "^5 

3x—2=^2x=— ix- _)2=4x* -2x3 _ ^2 4._^.^ J__^ ac- 
cording to the method in Case 2. 

5 , 25 5 

Then we shall have 3x — 2=-— x+-— ;, or 3x— ■— x=2' 

io 25b 15 

25 
-j — ;--. Whence, 768x-80x=5 12+25 ; and, consequently, 

512 + 25 537 



768—80 688 
Of, if we put x=-, the formula in that case will become 

y 

2{'M j^ y^ y 



218 DIOPHANTINE ANALYSIS. 

And, tberefore, multiplying thisby ?/* , which is a squarsc, 
it will be 'i — 2y~-y'+3y^-2y*. Where the first term 
being now a square, if the expression, so transformed, be 

resolved by Case 1, we shall have y=-—- • and x=-= 
y . ^ bol y 

- — , asbeiore. 
b-88' 

3. It is required to find such values of x, as will make 
j+3a;4-7x3 — 2a;3 + 4x« a square. 

Here, both the first and last terms being squares, let 1-t- 

o 25 

3a;-|-7x2-2a;3+4x* = (H-|x'+2x3)a = l+3x4-— a:?4-6 

:c^-\-4x*, according to the method in Case 3. 

'" 5 
Then, we shall have 6x3+— x2 =7x3 —2x3 ; or6x+2 

25 3 

x=7 —; and, by reduction, x=—. 

And, if we put the same formula l+3x+'ix2—2x^-{- 

4a:4=:(l-f|.T— 2x2)*,=l+3x-Jx2— 6a:3+4x*, we shalS 

have, by cancelling, Tx^ -2x= = -fx3— Gx^ ; whence Cx 

35 35 

.^2x=_i-7 = --;orx---^. 

And, in a similar manner, other values of x maybe found, 
by employing the method of substitution pointed out in the 
latter part of Case 3. 

4. It is required to find such values of x as will make 
2x* — 1 a square. 

Here, 1 beins; an obvious value of x, let, according to 

^'Then 2x^-1-^2(1+2/)* ~l=2(l + 4t/+62/=^ +43+2/*) 
_l = l + 82/+12j/2+8i/3 + 2y^. And since the first term 
of this last expression is novv a square, we shall have, by 
Case 1, l+8i/+12y3+8i/3 + 22/* = (l+42/-22/-)^ = l+% 
+ 12y2_i6(/3 + 4t/*» 

Whence, as the three first terms of the two numbers of 
this equation destroy each other, there will remain 4?/* — 
]G^''=2i/''+8j/'* ; OTy~\2; and, consequently, x= J +a/ 
= 13 ; which value being substitsted for r, makes 2x* - I 



urOPHANTLVE ANALYSIS. 219 

=57121 =(239)2, as required. And if 13 ba now taken, 

as the known value of .r, and the operation be repeated as 

before, we shall obtain, for another value of x, the conrfTli 

,^ . 1U607469769 
cated fraction ^^--^. 

PROBLEM IV. 

To find such values of x as will make ^{ax^ -fhx^-\-cx 
-\-d) rational, or ax^+iar^+cx+d— a cube.' This formula, 
like the two latter of those relating to squares,cannot be re- 
solved by any direct method, except in the cases where 
the first or last terms of the expression are cubes ; it being 
necessary, in all the rest, that some simple number answer- 
ing the conditions of the question, should be first found b^ 
trial, before we can hope to obtain others ; but when this 
can be done, the problem, in each of the cases here men- 
tioned, may be resolved as follows. 

RULE. 

I . When the last term d of the given formula is a cube, 
put it =e3, and xa'dkQe^-\-cx-\-bx^-\-ax^=^{e-\-—~xY -' 



- Then, by expunging the two first ternas on eack side of 
the equation, which are common, there will remain ax'-f- 

6x2= x^A x2 ; whence, by division and reduction, 

27e6 3e3 ' 

we shall have 27a€6x-f276c6=c3x-f-9c3e3^ and consequent- 
ly x= ^: ; which form fails when the coeffi- 

•^ c3— 27ae6 ' 

cients b and c, or a and c, are each equal 0. 

2. When the coefficient a of the first term is a cube, 

fc put it =/3, and va:ikepx^-\-hx^-^cx-\-d—(Jx-\--^^y ~ 



220 DIOPHANTINE ANALYSIS. 

Then, by expunging the two first terms on each side of 

h- 
tae eqLuatiou, as before, there will remain cx-\-d= -r- x-f 

—^ ; whence, by multiplying by 27/"«, we shall have 27 
f^cx-\-21df^=9b'^f^x-]-b^, and consequeetly a:=- 

9?V"~7 3--Z»2\ ' ^^"^^ ^°^"^ likewise fails, when h and c, 
or b and fii, are each =0. 

3. When the first and last terms are both cubes, put a= 
/2 and rf=e', and make e^ -\-cx-i-bx^ -{-f^x' = (e-\-fx)'^ =■ 
e3+3/e2.x-|-3/'2ea:2 +/3x3. 

Then, cx-i-bx'^=Sfe^x-\-3f'ex^ ; 
.Whence, we shall have bx—Sfsex=3Je^ — c ; and, conse- 

o /* o 

quently x= r~^pr 5 which formula may be resolved by 

either of the two first cases. 

4. When neither the first nor the last terms are cubes, 
let p be a value of x, found by inspection, or by trials, and 
make op^-\-bp- -{-cp-\-d=y^ . 

Then, by putting x=y-\-p, we shall have ap^-\-bp^-\-rp 
J^d=a{y^py+b {y^py+c(y+p)+d =«2/3+(3ap+A) 
1/2+ {Sap" A-2bp-}-c)y^ap^+bp^ -\-cp-\-d, or ax-^+bx^-f 
cx+d=ay^ + {3ap-\-b)y^-{-(3ap2+2bp+c)y-i-g^. 

From which latter form, the value of y, and conse- 
quently that of X, may be found, as in Case 1. 

EXAMPLES. 

1. It is required to find such a value of x as will make 
x^-^x-\- J a cube. 

Here, the last term being a cube, let the root of the 
cube sought — l-f-^a;, according to Case 1. 

Then, by cubing, we shall have l+x+ic^r^l-f-x-f-ix^ 

* 2 1 J 

And, since the two first terms on each side of this equa- 
tion destroy each other, there will remain x^=^^x^ -^■^\x''. 

Whence, dividing by x^, we shall have ^'yX+i=1, or 
:t4-9=27 ; and consequently x-=27--9=18 ; which num- 



DIOPHANTINE ANALYSIS. 221 

b.-r, by substitution, makes l+a;+a;2 — 1 + 18+324=343 
=7=* a cube number, as was required. 

And if we now take this value of a:, and proceed accord- 
ing to the method employed in Case 4, we shall obtains = — 

137826 , , . , •,,,,:,• ,1 

.^^ - ; which last number will also lead in like man- 
60653 

ner, to other new values. 

2. It is required to find such a value of x, as will make 
a;3+3x2 + 133 a cube. 

Here, the first term being a cube, let its root ^=-\-\-x., 
according to Case 2. 

Then, by cubing, we shall have 133+3x^+13 =(!--!-. 
= l + 3a;+3a;2+x3. 

And since the two last terms of this equation destroy 

each other, there will remain l+3x=13o, or 3x=133 — 

132 
1 = 132 ; whence x= =44, and x^ +3x2 + 133=9 11 25 

^^(45)=*, a cube number, as was re<^uired. 

And if 45 be now taken as a known value of x, other 
values of it may be found, as in the last example. 

3. It is required to find such a value of x, as will make 
3+28x+89x2 — 1:^5x3 a cube. 

Here, let the root sought =:l2 — 5x according to Case 3' 

Then, by cubing we shaM have 8+28x+89x2 — 125x=^ 
= (2-5x)3=8-C0x+150x3- 125x3. 

And, since the first and last terms of this equation ties- 
ti-oy each other, t-here will remain 28x+89x2= — 60x-}- 
150x2. 

Whence, by dividing by x, and transposing the terms we 

shall have 150x— 89x=28+C0, or 6ax=^8 ; and conse- 

88 
quently x=— . 
bl 

And as this formula can also be resolved either by the 

first or second case, other values of x, may be obtained, 

that will equally answer the conditions of the question, 

4. It is required to find such a value of x, as will make 
2x' — 3x+7 a cube. 

Here, —1 being a value of x that is r-eadily foand, by 
inspection, let x=3/— !, agreeably to Case 4o 



222 DIOPHANTINE ANALYSIS. 

Then, by substitution, we shall have 2x3^3a;+7=2(r/ 

And as the last term of this expression is a cube, let 8 
+3y-6if-\~2y^={2+^yy = 8-^3y+-y^+—y^ accord- 
ing to Case 1. Then, by expunging the equal terms on 

3 , 1 
each side, there will remam 2y^ — Gy^ "^Till^ ""fil^'' 

Whence, dividing by j/^, and reducing the terras, we 
shall have 128;/ — 384 = i/+24, or 127i/=408 ; and conse- 

408 , 408 , 281 

quently y— , and x = — --— l=T^r:;- 

4 jy J27' ,27 127 

Which number, by substitution, makes 2x' — 3x4-7= 

2X(28n3 281 , 46118016 /356n3 

. — ^ <- 3X +7= =( - — ) , as requir- 

(127)3 127^ 2048383 \ 127/ ' ^ 

ed. And, by taking this last as a new value of x, others 

may be determined by the same method. 



PROBLEM V. 

Of the resolution of doubl^nd triple equalities. 

When a single formula, containing one or more unknown 
quantities, is to be transformed into a perfect power, 
such as a square or a cube, this is called, in the Diophan- 
tine Analysis, a simple equality ; and when two formulae, 
containing the same unknown quantity, or quantities, are 
1o be each transformed to some perfect power, it is then 
called a double equality, and so on ; the methods of resolv- 
ung which, in sued cases as admit of any direct rule are as 
follows ; 

■RULE. 

S. In the case where the unknown quantity does nx)t 
exceed the first degree, as in the double equality 



DIOPHANTINE ANALYSIS. 225 

ax-\-b—0,BTi(i cx-Td=Q, 
let the first of these formulas ax-}-b=2^, and the second 
.€x-\-d=w^. 

Then, by equating the two values of x, as found from 
these equations, we shall have cz^ -\-ad — bc=aw^ , or acz^ 
-\-a(ad—bc)~a^w^. 

And since the quantity on the right hand of this equation 
is now a square, it only remains to find such a value of z 
as will make, when the question is resolvable, acz^'\-a(^ad 
,-^bc) = n; which being done, according to the method 

pointed out in Problem 1, we shall have x= . 

2. When the unknown quantity does not exceed the 
second degree, and is found in each of the terms of the 
two formulae ; as in the double equality 

0x2+60,== Di and cx^-\-dx = 0. 
Let x=|, then, by substitution, and multiplying each of 
the resulting expressions by 7j~ , we shall have 

a-\-by=*a , a»d c-}'dy= Q , 
from which last formula, the value ofy, when the question 
is possible, and consequently that of x, may be determined 
as in Case 1 . 

But if it were required to make the two general expres- 
sions 

ax* +6.T+c= D , and dx^ +ex-j-/= D , 
the solution could only be obtained in a few particular 
cases, as the resulting equality would rise to the fourth 
power. 

3. In the case of a triple equality, where it is required 
to make 

ax-{-hy= C i cx-^dy'= Q , and ex+/?/=n , 
let the first of them ax-\-by=u" , the second cx+dy=v^, 
and the third ex-\-fy=w^ . 

Then, by first eliminating x in each of these equa- 
tions, and afterwards y, in the two resulting equations, we 
shallhave {af—be)v^ -{cf-de)u^={ad-bc)vc'> 
■or, putting r=«r, and reducicg the terms, the result will 

af — be cf — de w^ 

jgive the simple equahtv —^ — -z^ — - ——^ ; "wnere 

* ^ ^ - ad-~bo ad — be m^ 



224 DIOPHANTINE ANALYSIS. 

the right hand member being a square, it only remains to 

find a value of z that will make the left hand member a 

square ; which, when possible, may be done by Problem 1, 

Hence, having z. we have as above, v=siuz ; and the first 

.„ . d~ bz^ , az^ — c 

two equations will givex=— ; — -r-«^» and y=- — ; ;-"", 

^ *= ad —be ^ ad —be 

"where u may be any whole or fractional number whatever. 

But if the three formulee, here proposed, contained only 
one variable quantity, the simple equality, to which it 
would be necessary to reduce them, would rise, as in the 
last case, to the fourth power ; and be equally limited 
with respect to its solution. 

4. In other cases of this kind, ail that can be done is to 
find successively by the former rules, several answers, 
when one is known ; and, if neither this nor any of the 
above mentioned modes of solution are found to succeed, 
the Problem under consideration can only be determined 
by adopting some artifice of substitution that will fulfil one 
or more of the required condition*, and then resolving the 
remaining formulae, when they are possible, by the me- 
thods already deUvered for that puppose ; but as no gene- 
ral precepts can be given, for obtaining the solution in this 
way, the proper mode of proceeding, in such cases, must 
chiefly depend upon the skill and sagacity of the learner. 

EXAMPLES. 

1. It is required to find a unmber x, such that a. -{-1 28 
and a;4-192 shall be both squares. 

Here, according to Case l,letx-}-\28=^w^, and a;-j-192 

Then, by eliminating z, and equating the result, we 
shall have w^- 128=23- 192, or w''-{-e4=z-. 

And, as the quantity on the right hand side of the equation 
ES now a square, it only remains to make w^ -\-64 a square. 

For which purpose, put its root =w-\-n ; then ii)2 4-64 
=a>' -^2nw-{-n^ , or 2nw-\-n' =C4 ; and consequently w = 

— - — ; where taking n, wliich is arbitrary, =2, we Bhall 



DIOPHANTINE ANALYSIS. £25 

'have w= — ^^=^-1=15 ; and consequently x=^W — 128 
4 4 

= 132 — 128=225— 128=97, the answer. 

2. It is required to find a number a;, such that x^-f-x 
and x^ - X shall be both squares. 

Here, according to Case 2, of the last Problem, let x= 

1 ,1.1,11 

- : then we shall have to make— -i — , and— - — -squares ; 

:y r y y y 

or, by reduction, — ( 1 -\-y) — D > and— ( 1 — 2/) = □ • 

Or, since a sqiare number, when divided by a square 
■Jiumber, is still a square, it is the same as to make 

1+^=D and 1 -i/=n, 
-for this purpose, therefore, let l+2/=-2^» or 7/=^- — -1 i 
then 1 —y—'i—z^ ; which is also to be made a square. 

But as neither the first nor last terms of this formula, 
,are squares, we must, in order to succeed, find some sim- 
ple number, that will answer the condition required ; 
■which, it is evident, from inspection, will be the case 
whenzr=i. 

Let, therefore, z—\ -tw, agreeably to Problem 1, Case 
7, and we shall have 1—^=2— 22— 2._(i_a;)? = 14-2w 

— :ii)2 ; or 2/^=^^ —2a' ; 

.or, putting X—n-w for the root of the former of these ex- 
pressions, there will arise, by squaring, \-\-tisO'~m'^^^\^- 
2ntt'4-n2tiy2. 

Whence, expunging the one on each side, and dividing by 
ty, we shall have 2~a;=— 2n+n2t0 ; and consequently 



^n 



f-2 - 1 1 (?(24-l)2 



xio=- ; — , and .r=- 



?i3-[_r y «)3— 2ay 4rt— 4n^ 

where, in order to render the value of x positive, n may 
be taken equal to any proper fraction whatever. 



rii 



Or, if for the sake of grenter generality, — be substi- 

n 

toted for », we shall then have 

3;^^ i L 



226 DIOPHANTINE ANALYSIS. 

where m and n may now be taken equal to any integral nuiii 
bers whatever, provided n be made greater than m. 

or 
If, for instance, n=2 and m=l, we shall have a;=— 

169 
and if n=3 and 7ft=2, ^=-r^7: ; and so on, for any ot]i: < 

number. 

3. It is required to find three whole numbers in arithme- 
tical progression, such, that the sum of every two of the r; 
should be a square. 

Let X, x-ry, and x+2y, be the three numbers sought ; 
and put 2x-\-y=u^,^x-{-2'j=v^, and 2x-f-3y=a'2, agreea- 
bly to Case 3. 

Then, by eliminating z and y from each of these equa- 
tions, we shall have v^ -u^=^w^ ~t>^ , or 2x)2 _ u- =w^ . 

And, if we now put ■y=u2', there will arise 2u^z^ —1*2 = 

W ; or by dividing by u^,^z'' — l—— ; where the right 

hand member being a square, it only remains to make 2z^ 
— 1 a square, which it evidently is when z= 1 . 

But as this value would be found not to answer the con- 
ditions of the question, let z=l — p ; then 2z^ — 1=2(1 — 
p)'-l = l— 4p-f2/j2. 

And, consequently, if this last expression be put =(1 — 

np)", we shall have, by squaring, 1 — 4^+2^2 = 1 _2np+ 

n^p^y or ^4-^2p = — 2n-\-n^ p ; whence 

2»-4 - , 271 — 4 n^—2n-\-2 

p= and i=l 



?i2 _ 2 ?i3 _ ; 



n- 



or, if, for the sake of greater generaUty, — be substituted 

n 

for n in this last expression, we shall have 
_m2 -2?7in-}-2ji2 

And since, by the two first equations, y=zv^ —■us — ^2 -a 
— u^=(^z2 _l)«2,and x^^{n^ -y)=-}{2-z'')u^. iiis evi- 
dent, that 2 must be some number greater than >, and less 
than ^s!. 

If, therefore, m=^9 and w--=5 we shall have 



DIOPHANTINE ANALYSIS. 227 

_C1 -90+50 41 241 u2 709 

' 8Ti:50~==3T'^=3F><T""^ 2/=3^I^^'" 

Or, taking «=2X31,a;=482, and j/=2880, we have x= 
182, x4-2/=3362, and a;+27/=6242, which are the num- 
Sers required. 

4. It is required to divide a given square number into 
wo such parts, that each of them shall be a square.* 

Let «2=given square number, and x- and oJ^ ~ x^ its 
wo parts. Then since x^ is af juare, i^only remains to 
nake a- -x^ a square. 

For which purpose let its root —nx—a and we shall have 
*' — a;3=n3x3— 2a/?a:-f«3,or— x2=n2x3— 2a72x;whence, 

y reduction, x= the root of the first part : and ux 

_ 2an2 an2 — a 

^— ~rTT~"~ ■> I r *'^6 J^oot of the second. 

Therefore(^^^J^^''and('^^^^-^y are the parts re- 

uired ; where a and n may be 'any numbers taken at 
leasure, provided n be greater than 1 . 
5. It is required to divide a given number, consisting of 
i'o known square numbers, into two other square num- 
;rs. 

Let a'^-\-h- be the given numbers, and x^, y2 the two 
quired numbers, whose sum, x^-fija, is to be equal to 

+62. ^ ^ 

Then it is evident, that if x be either greater or less 
an o, ?/ will be accordingly less or greater than h. Let 
erefore x=a+j>jz, and y=^—nz, and we shall have a^ 

Or, by transposition and rejecting the terms which are 

To this we may ad-i the following useful property, 
f s and r be any two unequal numbers, of which s is the greater, it ca« 
a be readily shown, from th^ nature of the problem, that 

2rs, $2 — >-2 and sn + '"2 
1 be the perpendicular base, and hypothenuse of a right-angled triangle, 
'rom which expressions, two square numbers may be found, whose sum 
difference shull be square numbers; for {2rs)2 +.(*2 — r3)a =(^2 + 
2, and (s2 -}-»-2)2 — (2r«)3 =(52 — »-2)2, or (.s-2 +r3)2 — (jS — rr)« 
'2r-;2 ; where s and >• may be any numbers whatevef. 



228 DIOPHANTINE ANALYSIS. 

common to each side of the equation, m^z^'{'n^2^=^inz 
.^2atm, or m2 2-fn2z=26n - 2am ; whence 

where m and n may be any numbers, taken at pleasure, 
provided their assumed values be such as will render the- 
values of x, y, and z, in the above expressions, all posi-;' 

6. It is required to find two square numbers, such that' 
their difference shall be equal to a given number. 

Let d= the given difference ; which resolve into two ; 
factors a. b ; making a the greater and b the less. ; 

Then, putting x= the side of the less square, and x-j-.. 
b= side' of the greater, we shall have (a:+6)2 — x^ =x2" 
^2bx+b''-x^=cl{ab) or 2bx+b^=d{ab). 

Whence, dividing each side of this equation by b, wei 

shall have.x=^^= the side of the less square sought, 

and a;+6=^+6=^= the side of the greater. 

If, for instance, d—6L\ take aX?»=30X2, and we shall 

have x=^-^-14, and ..+2=^=16, or 16^-14= 

2 ■^ 

=256 ^ 196=30 the given difference. 

7. As arj instance of ihe great use of resolving formula 
of this kind into factors, let it be proposed in addition to 
what has been before said, to find two numbers x and y, 
such that the difference of their squares, x^—y- , shall btf 
an integral square. 

Here the fo.'tors of x^ —y^ , being x+y and x - y. we 
shall have (x+t/) X {x-y)=x^ - y"" ■ And, since this pro- 
duct is to be a squ^^re, it will evidently become so, by mak- 
ing each of its factors a square, or the same multiple of | 

square. i 

Let there be taken, therefore, for this purpose, j 

x-\-y=niz^, X — y=^ms^. 
Then, by the question, we shall have (x+i/)X(x-t^5 
or itsj equal rs ..y^=m''f^s^; which is evidently a square 
whatever may be the values of m, r, s. ' 



DIOPHANTINfi ANALYSIS. 229- 

But, by addition and subtraction, the above equations 
give, when properly reduced 

^"""2 ' y^ 2 ' 

where, as above said, m, r, and s, may be assumed at plea- 
sure. Thus, if we take m=2, we shall have x=r^-{-s^, 
and y=r^—s-, which expressions will obviously give in- 
tegral values of x and y, if r and s be taken = any integral 
numbers. 

8. It is required to find two numbers, such that, if ei- 
ther of them be added to the square of the other,^ the 
"sums shall be squares. 

Let X and y be the numbers sought ; and consequently 
x' +i/ and y^ -{-x the expressions that are to be transformed 
into squares. Then ifr — x be assumed for the side of the 
first square, we shall have x^+y=r' — 2rx-\-x^ , or ?/='"'' 

r^ — y 
— 2rx ; and consequently x= -^ ^ . 

And if s-\-y be taken for the side of the second square, 

we shall have 2/2 -j — ^_£.=:s2.^2s?/-}.j/2 ; or, by reducing 

the equation, r^ —y=Arsy-\-'ir$^ , and consequently, by re- 

,.2— 2rs2 2r=^s-fs2 

duction, y—-— — -tt—, ana ^——^ — TT" 5 where r and s 
"^ 4rs-fl 4r5-fl 

tnay be any numbers, taken at pleasure, provided r be great- 
er than 2s2. 

9. It is required to find two uMnbers, such that their sum 
and difference shall be both squares. 

Let X and x^ —a; be the two numbers sought ; then, since 
their sum is evidently a square, it only remains to make 
their difference, .r^— 2a;, a square. 

For this purpose, therefore, put its root =a; — r and we 
shall have x^ -^'J;x=x^ — 2rx + r" ; 

Or, by transposition, and cancelling a;^ on each side of 
tke equation, 2rx — 2a;=r2 ; whence 



^'=2£:2' ^"'^ ^'-^^^K^— i) 



r-l' 



230 DIOPHANTINE ANALYSIS. 

where r maybe any number, taken at pleasure, provided 

it be greater than 2. 

10. It is required to find three numbers, such that not 

only the sum of all three of them, but also the sum of 

every two shall be a square number. 

Let 4x, x2 —4a; and 2x-i- 1, be the three numbers s©ugbt, 

then4a + (x2 -4a;)=x-2, (x^— 4x)-f (2x+l)=x2— Zx+l, 

and 4^+(x2-4x)-|-(2x-|- l)=x -}-2x+ 1 , being all squares, 

it only remains to make 4x4-(2x+l), orits equal, 6x+l,a 

square. For which purpose, let 6x-\-l=n*, and we shall 

n^ — l 
have, by transposition and division, x= — :; — , whence, 

6 

4n2 — 4 (n3_l) 4n^-~4 , 2«2— 2 . . . 

—6--' 36" ' 6—' «°d-g— +1, or their e- 

, 2n2_2 n*— 26n2+25 , n^-{-2 

quals — - — , —^ , and — - — are the numbers 

o ob 3 

required. 

Where n may be any number, taken at pleasure, pro- 
vided it be greater than 5. 

QUESTIONS FOR FRACTICE. 

1. It is required to find a number x, such that x+l and 
X— 1 shall be both squares. Ans. x=|. 

2. It is required to find a number x, such that x-\-4 and 
x+T shall be both squares. Ans. |-|. 

3. It is required to find a number x, such that lU-f-sc 
and 10 — X shall be both squares. Ans. x=6. 

4. It is required to find a number x, such thatx^-f-l 
and x+1 shall be both squares. Ans. Y* 

5. It is required to find three integral square numbers, 
-such that the sum of every two of them shall be squares, 

Ans. 628, 5796, and 6325. 

6. It is required to find two numbers x and y, such that 
x2 +2/ and 2/2 4-x shall be both squares, 

Ans. x=WV, and i/=t\* 

7. It is required to find three integral square numbers, 
that shall be in barmonical proportion. 

Ans. 25, 49, and 1225. 



DIOPHANTINE ANALYSIS. 23 i 

8. It is required to 6nd three integral cube numbers, 
x'^, y^, and z^^ whose sum may be equal to a cube. 

Ans. 3, 4, and 5. 

9. It is required to divide a given square number (100) 
into two such parts that each of them may be a square 
number. Ans. 64, and 36. 

10. It is recfuired to find two numbers, such that their 
difference may be equal to the difference of their squares, 
and that the sum of their squares shall be a square num- 
ber. Ans. 4 and ^. 

11. To find two numbers such, that if each of them be 
added to their product, the sums shall be both squares. 

Ans. I and |. 

12. To find three square numbers in arithmetical pro- 
gression. Ans. 1, 25, and 49 

13. To find three numbers in arithmetical progression 
such that the sum of every two of them shall be a square 
number. Ans. 120i, 840^, and 1560i. 

14. To find three numbers such, that if to the square 
of each the sum of the other two be added, the three 
sums shall be all squares. Ans. 1 , f and ^ 

15. To find two numbers in proportion as 8 is to 15, 
and such that the sum of their squares shall be a square 
number. Ans. 576 and 1080 

i6. To find two nurabers such, that if the square of 
each be added to their product the sums shall be both 
squares. Ans. 9 and 16. 

17. To find two whole numbers such, that the sum or 
difference of their squares, when diminished by unity shall 
be a square. Ans. 8 and 9 

18. It is required to resolve 4225, which is the square 
of 65, into two other integral squares. 

Ans. 2704 and 1521 

19. To find three numbers in geometrical proportion 
such that each of them, when increased by a given num- 
ber (19), shall be square numbers. Ans. 81, f and ^ff„ 

20. To find two numbers such, that if their product be 
added to the sum of their squares, the result shall be a 
square number. Ans. 5, and 3, 8 and 7, 16 and 5, &c. 



232 DIOPHANTINE ANALYSIS. 

21. To find three whole numbers such, that if to the 
square of each the product of the other two be added, the 
three sums shall be all squares. Ads. 9, 73, and 328 

22. To find three square numbers such that their sum 
wh-en added to each of their three sides, shall be all square 
Bunabers, 

Ans. eVir'/o. iUU-- i^nd i|||i=roots required 

23. To find three numbers in geometrical progression 
such, that if the mean be added to each of the extremes, 
the sums, in both cases shall be squares. 

Ans. 6, 20, and 80 

24. To find two numbers such, that not only each of 
them, but also their sum and their differerce, when in- 
i"rea$ed by unity, shall be aH square numbers. 

Ans. 3024 and 5624 

25. To find three numbers such, that whether their sum 
be added to, or subtracted from, the square of each of 
them, the numbers thence arising shall be all squares. 

Ane 40.6 5_i_8 and V-' 

26. J o find three square numbers such, that the sum of 
their squares shall also be a square number. 

Ans. 9, 16, and V/ 

27. To find three square numbers such, that the differ- 
ence of every two of tJiem shall be a square number. 

Ans. 485809, 34225, and 23409 

28. To divide any given cube number (8) into three 
other cube numbers, Ans. 1, || and VV 

29. To find three square numbers such, that the dif- 
ference between every two of them and the third shall 
be a square number. Ans. 1492, 24 P, and 2692 

30. To find three cube numbers such, that if from each 
of them a given number (1) be subtracted, the sum of the 
remainders shall be a square number. 



(233) 



OF THE 



SUMMATION AND INTERPOLATION OF 
INFINITE SERIES. 

The doctrine of Infinite Series is a subject which has 
engaged the attention of the greatest mathemaficians, both 
of ancient and modern times ; and, when taken in its 
whole extent, is, perhaps, one of the most abstruse and 
difficult branches of abstract mathematics. 

To find the sum of a series, the number of the terms of 
which is inexhaustible, or infinite, has been regarded by 
some, as a paradox, or a thing impossible to be done ; but 
this difficulty will be easily removed, by considering that 
every finite magnitude whatever is divisible in infinitum^ 
or consists of an indefinite number of parts, the aggregate, 
or sum, of which, is equal to the quantity first proposed. 

A number actually infinite is, indeed, a plain contradic- 
tion to all our ideas ; for any number that we can possibly 
conceive, or of which we have any notion, must always 
be determinate and finite ; so that a greater may still be 
assigned, and a greater after this ; and so on, without a 
possibility of ever «oming to an end of the increase or ad- 
dition. 

This inexhaustibility, therefore, in the nature of num- 
bers, is all that we can distinctly comprehend by their in- 
finity ; for though we can easily conceive that a finite 
quantity may become greater and greater without end, yet 
we are not, by that means, enabled to form any notion of 
the ultimatum, or last magnitude, which is incapable of 
farther augmentation. 

Hence, we cannot apply to an infinite series the com- 
mon notion of a sum, or of a collection of several particu- 
lar numbers, which are joined and added together, one 
after another ; as this supposes that each of the numbers 
composing that sura, is known and determined. But as 
every series generally observes some regular law, aji4 

X 2 



234 SUMMATION OF INFINITE SERIES. 

continually approaches- towards a term, or limit, we can 
eapily conceive it to be a whole of its own kind, and that 
it must have a certain real value, whether that value be 
determinable or not. 

Thus in many series, a number is assignable, beyond 
which no number of its terms can ever reach, or indeed, 
be ever perfectly equal to it ; but yet may approach to- 
wards it in such a manner, as to differ from it by less than 
any quantity that can be named. So that we may justly 
call this the value or sum of the series ; not as being a 
number found by the common method of addition, but such 
a limitation of the value of the series, taken in all its infi- 
nite capacity, that,, if it were possible to add all the terms 
together, one after another, the sum would be equal to that 
number. 

In other series, on the contrary, the aggregate, or value 
of the several terms, taken collectively, has no limitation ; 
which state of it may be expressed by saying, that the sum 
of the series is infinitely great ; or, that it has no deter- 
minate or assignable value, but may be carried on to such 
a length, that i\M sum shall exceed any given number what- 
ever. 

Thus, as an illustration of the first of these cases, it 
may be observed, that if r be the ratio, g the greatest 
term, aod / the least, of any decreasing geometric series, 
the sum, according to the common rule, will be (rg—l)-^ 
(r — 1) : and if we suppose the less extreme, I, to be di- 
minished till it becomes =0, the sum of the whole series 
will be rg^{r — l) : for it is demonstrable, that the sum 
of no assignable number of terms of the series can ever 
be equal to that quotient ; and yet no number less than it 
will ever be equal to the value of the series. 

Whatever consequences, therefore, follow from the sup- 
position of rg-T-{r— 1) being the true and adequate valae 
of the series, taken in all its infinite capacity, as if all the 
parts were actually determined, and added together, no as- 
signable error can possibly arise from them, in any ope- 
ration or demonstration where the sum is used in that sense ; 
because, if it should be said that the series exceeds that 



SUMMATION OF INFINITE SERIES. 235 

value ; it can be proved, that this excess must be less than 
any assignable difference ; which is, in effect, no differ- 
ance at all ; whence the supposed error cannot exist, and 
consequently rg-^^r- 1) may be looked ujion as express- 
ing the true value of the series, continued to infinity. 

We are, also, farther satisfied of the reasonableness of 
this doctrine, by finding, in fact, that a finite quantity is 
frequently convertible into an infinite series, as appears 
in the case of circulating decimals. Thus two-thirds ex- 
pressed decimally is |=.66666, &c. = j% + yf^ + r/oo 
_j_ y^f p^o + &c. continued ad infinitum. But this is a 
geometric series, the first term of which is y\, and the 
ratio y'o ; and therefore the sum of all its terms, conti- 
nued to infinity, will evidently be equal to |, or the num- 
ber from which it was originally derived. And the same 
may be shown of many other series, and of all circulating 
decimals in general. 

With respect to the processes by which the summation 
of various kinds of infinite series are usually obtained, 
one of the principal is by the method of differences point- 
ed out and illustrated in Prob. iv. next following. 

Another method is that first employed by James and 
John Bernoulli, which consists in resolving the given se- 
ries into several others of which the summation is known ; 
or by subtracting from an assumed series,when put =s, the 
same series, deprived of some of its first terms ; in which 
case a new series will arise, the sum of which will be 
known. 

A third method which is that of Demoivre, consists in 
putting the sura of the series =s, and multiplying each 
side of the equation by some binomial or trinomial expres- 
sion, which involves the powers of the unknown quantity 
X, and certain known co-efficients ; then taking x, after 
the process is performed, of such a vfilue that the assum- 
ed binomial, &c. shall become =0, and transposing some 
of the first terms, a series will arise, the sum of which will 
be known, as before. 

Each of which methods, modified so as to render it 
more commodious in practice, together with several other 



236 SUMMATION OF INFINITE SERIES. 

artifices for the same purpose, will be found sufficiently 
elucidated in the miscellaneous questions succeeding the 
following problems. 

PROBLEM I. 

Any series being given to find its several orders of dif- 
ferences. 

RULE. 

1. Take the first term from the second, the second from 
the third, the third from the fourth, &.c. and the remainders 
will form a new series, called the Jirst order of differences. 

2. Take the first term of this last series from the se- 
cond, the second from the third, the third from the fourth, 
&c. and the remainders will form another new series, call- 
ed the second order of differences . 

3. Proceed in the same manner, for the third, fourth, 
fifth, &c. orders of differences ; and so on till they termi- 
nate, or are carried as far as may be thought necessary*. 

EXAMPLES. 

L Required the several orders of differences of the se- 
ries 1, 22, 3S 43,52, 62, &c. 
1, 4, 9, IC, 25, 36, &c. 

3, 5, 7, 9, II, &c. 1st diff. 
2, 2, 2, 2, &c. 2d diff. 
0, 0, 0, &c. 3d diff. 
2. Required the several orders of differences of the se- 
ries 1, 23, 33, 43, 53, 63, &c. 

1, 8, 27, 64, 125, 216, &c. 

7, 19, 37, 61, 91, &c. 1st diff. 

12, 18, 24, 30, &c. 2d diff. 

6, 6, 6, &c. 3d diff. 

0, 0, &c. 4th diff. 



* When the several terms of the series continually increase, the diflerencei 
will be all positive ; but when they decrease, the diifereuccs nill be jicgative 
and positive alternately. 



SUMMATION OF INFINITE SERIES. 237 

3. Required the several orders of diflferences of the se- 
.Ties 1, 3, 6, 10, 15, 21, Lc. 

Ans. 1st, 2, 3, 4, 5, &c. ; 2d, 1, 1, 1, kc. 

4. Required the several orders of differences of the se- 
ries 1, 6, 20, 50, 105, 196, &c. 

Ans. 1st, 5, 14, 30, 55, 91, &;c. ; 2d, 9, 16, 25, 
^^ 36, &c. ; 3d, 7, 9, H, &c. ; 4th, 2, 2, &c. 

Wf' S. Required the several orders of differences of the 
.11111. 
*^"^^2'i'8'T^'32'^^- 



► 



rROELEM II. 



Any series «, fe, c, d, e, &c. being given, to find the 
first term of the nth order of differences. 

RULE. 

Let S' stand for the first term of the »th differences. 

^, .„ , , n— 1 n—l n— 2 ?i— 1 

Then will a — 7ib -f- n.———c — 7i.—~—.——a-\-n.—--' 

n— 2 n— 3 



e, &c.to n+1 terms =(J*, when Ji is an even 



3 4 

number. 

n—l , n— In— 2 n— In— 2 

■ And — fl + n6— n.-^e+n.— ^.—^a-n.-^.-^. 

n-3 



e .Sic. to n -{- 1 terms =<5", when n is an odd num- 



3 
ber.* 

EXAMPLES. 

1. Required the first term of the third order of differ- 
ences of the series I, 5, 15, 35, 70, hz. 

Here a, 6, e, rf, e, &.c. =«:1, 5, 15, 35, 70, &c and n=3. 

\a hence — a -\- 1^*^ — f^'—^ — c-\-n. —- — . — - — a = — a 



* When the terms of the several orders of differences happen to be verv 
■great, it will be more convenient to take the logarithms of tlie quantities con- 
cerned whose differences will be smaller ; aud when the operation is tiaijhetis 
the (quantity answering to the last logarithm may be easily found. 



238 SUMMATION OF INFINITE SERIES. 

4-36—3c+£i= — 14-15— 464-35=4= the first term re- 
quired. 

S. Required the first term of the fourth order of differ- 
ences of the series 1, 8, 27, 64, 125, &c. 

Here a, b, c, d, e, &c. =1, 8, 27, 64, 125, &c. and »=4. ^ 

•.XT, ,, n-l n— Iti— 2,, n— 1 

Whence a — nb-{-n. c~-n. . d+n. 

2 2 3 ^ 2 , 

~^.—^e=a-4b+ec- 4c/+e=l _32-f 162 - 2564-126 

?=0 ; so that the first term of the fourth order is 0. 

3. Required the first term of the eighth order of differ- 
ences of the series, 1, 3, 9, 27, 81, &c.* Ans. 256. 

4. Required the first term of the fifth order of difier- 

r.i, , 1 1 1 1 1 t 

ences of the series, 1, -, -, -, — , — , — , &c. 

' 2' 4' 8' 16' 32' 64' 

Ans. -— . 
ob 

PROBLEM III. 

To find the nth term of the series, a, b, c, d, c, &c. 
when the differences of any order become at last equal to 
each other. 

RULE. 

Let d', d", d'\ tZ'^, &c. be the first of fhe several orders 
©f differences, found as in the last problem. 

Ifaen will a-[ d'-\- . d'A . . 

^1 ^l 2^123 
J,., n — 1h — 2n — 3ra— 4 ,. ^ , 

" + — - — • o ' -i • — i — >""' *^c. =nth term required. 

1 ^ o 4 

EXAMPLES. 

1 . It is required to find the twelfth term of the series 
2, 6, 12, 20, 30, &c. 



* The labour, in quesiions of this kind may be often abridged, by putting 
ciphers for some of the terms at the beginning of the series ; by which means 
several of the ditlVrcnct s will be equal to 0, and the answer on that account, 
obtaiced La fetrer tertns. 



SUMMATION OF INFINITE SERIES. 23? 

2, 6, 12, 20, 30, &c. 

4, 6, 8, 10, &c. 

2, 2, 2, &c. 

0, 0, &c. 

Here 4 and 2 are the first terms of the differences. 

Let, therefore, 4=d', 2=d", and ?i=12. 

rr,, . ^ 1 « . 1"" 1 71— -2 „, . „ . 

Then a-\ d:+ ---_.__ d =2+ 1 Id ■+- .55d" = 2 

112 

+44+110=156=12th term, or the answer requu-ed. 

2. Required the twentieth term of the series, 1 , 3, 6, 
10, 15, 21, &c. 

1, 3, 6, 10, 15, 21, &c. 
2, 3, 4, 5, 6, &c. 

1, 1, 1, 1, &LC. 

0, 0, 0, &c. 
Here 2 and 1 are the first terms of the differences. 
Let, therefore, 2=d', l=d'', and n=20. 

Then 0+— d'+^^.^^d"= 1 4- 1 9(r +1 7 1 (Z" = 1 + 

38+171=210=20th terra required. 

3. Required the fifteenth term of the series, 1 , 4, 9, 16, 
.25,36, &c. Ans. 225. 

4. Required the twentieth term of the series, 1, 8, 27, 
64, 125, &c. , 

1 1 

5. Required the thirtieth term of the series, 1> g, ^, 

10' 15'21' ^- 

PROBLEM IV.* 

To find the sum of n terms of the series, a, i, c, d^ e, 
fee. when the differences of any order become at last 
equal to each other. 



• When the differences in this or the forn>er rule are finallj^=0, aay term, 
or the Slim of atiy number of the terms, may be accurately ^:termined ; but 
if the differences do not vanish, the result is only an approximation ; which, 
however, may be often very usefully applied in resolving various questions 
that may occur in this branch of the subject, and which will beconae conti* 
nually nearer the truth as the differences diminish. 



240 SUMMATION OF INFINITE SERIES. 



RULE. 



Let d', d'', d'", rf", &c. be the first of the several or- 
ders of differences.. 

n-i,,, n— 1 «— 2 71-1 

1 hen will na -f-n . a -t-n.—- — . — — — d -f-n 



2 ' 2 3 ' 2 

n— 2n — 3 „,. , n—ln—2n — 3n — 4,. 

.— —d'"-\-n.— —.-—-.-— —.——d'\kc. = to the 

4 2 3 4 6 

sum of n terms of the series. 



EXAMPLES. 

1. Required the sum of n terms of the series, 1, 2, 3, 

4, 5,-6, fee. 

Here 1, 2, 3, 4, 5, 6, &c. 

1, 1, 1, 1, 1, &.C. 

0, 0, 0, 0, &c. 

Where 1 and are the first terms of the differences, 

Let, therefore, a=l, d'=l, and d"=0. 

•II . ^t— 1 „ , n'^ —n n^-{-n 

Then will na-f-n.—-—d=n-\ — = — - — =sum of n 

2 2 2 

terms, as re^juired. 

2. Required the sum of « terms of the series j 1-, 22,33 J 

42, 52, &c., or 1, 4, 9, 16, 25, &.C. 

Here 1, 4, 9, 16, 25, &c. 

3, 6, 7, 9, &c. 

2, 2, 2, &c. 

0, 0, &ic.- 

Whcre 3 and 2 are fhe first terms of the differences, • 

Let therefore, a=l, d'=3, and d'=2. 

1 hen will na-f-n . — — a -f-n • — ^ — • — ^ — d =n-\-3n. 

n-l , n-1 «-2 3n2— 3fi , n3-3«2+2rt. 

--2- +2^--^ 3- = — 2--+ 3 = 

»x(ra+l)X(2n+l) . ^ . . 

— ^^ — ^^ ~ = sum of « terms as required. 



SUMMATION OF INFINITE SERIES. 241 

3. Required the sum of n terms of the series, P, 2', 
33, 43, 53, &,c., or 1, 8, 27, 64, 125, &c. 

Here 1, 8, 27, 64, 125, &c. 

7, 19, 37, 61, &c. 

12, 18, 24, &c. 

6, 6, &c. 

0, &c. 

Where the first terms of the differences are 7, 12, ami 

5. 

Let, therefore, a=l, d'=l, d"=12, and d'"=6. 

n~~l ,. , n— In — 2,,. , n-l 
Then will na + n . —x-d +n • —^-—^^ +" • ~2~ 

-— -. —-^d:' = n-\-7n.—- — |-12n.-- — + 6fi , 

34 2 2 o 

,1-1 71—2 n-3 In^—ln c * . vi . 

2 3 4^2^ 

ft^-6?i3 + lln2-.67i 4n Un^~\4.n Sn^ — 247t2+ 16/^ 

4 T"^ 4 ' 4 

71*— 6ra3-f nn2_6?i 7z*+27i3-|-n2 

J = = sum of 71 terms as 

~ 4 4 

required. 

4. Required the sum of n terms of the series, 2, 6, 12, 

. nX(n4-l)X(7i+2) 

20, 30, &c. Ans. — ^ — | — ^ ^ 

• 3 

6. Required the sum of n terms of the series, 1 , 3, 6, 

71 71+1 «+2 

10, 13, &c. Ans. i'-^— •— 3- 

6. Required the sum of n terms of the series, 1, 4, 10, 

71 7i-rl n-|-2 71+3 
20, -35, &c. Ans. _._^.-^.-^ 

7. Required the sum of n terms of the series H, 2*, 
3S 4S &c., or 1, 16, 81, 256, &c. 

. n^ 71* , 71^ n 

8. Required the sum of 71 terms of the series 1^, 2^, 3*, 

7i« . 7i5 571* n- 
*^5^&c. ^"'•-F+T+T2""T2 



242 SUMMATION OF INFINITE SERIES. 



PROBLEM V. 



The series a, b, c, d, e, fcc. being given, whose terms 
are an unit's distance from each other, to find any interme- 
diate term by interpolation. 



RULE. 

Let x be the distance of any term y, that is to be inter- 
polated, from the first term, and d', d', d", k.c. the terms 
of the difl'erences. 

Then will a-\-xd'4-x.- d'A-x. "" . d"'4~x.-^^ 

2 2 3^ 

x—2 x—3 ,. „ 

d'% &c. =y. 



EXAMPLES. 

1. Given the logarithmic sines of 1° O', 1° 1 , l^* 2' and 
1° 3', to find the log. sine of 1" 1' 40". 

Here 1*^0' 1"!' 1» 2' I" 3' 

Sines 8.2418553 8.2490332 8.2560943 8.26S0424 

71779 70611 69481 

— 1168 • —1130 

38 

Whence the first terms of the differences are 71779, — 

1168, and 38. 

Let, therefore, x=l'> l' 40"— 1° 0'=1'40"=1|= dis- 
tance of y, the term to be interpolated ; and ^'=71779, d'' 
= —1168, and rf''=38. 

X 1 , X 1 X 2 

Then will y=a-\-xd'-{-x.^-^d''-{-x. — — . d'''=a-i- 

~d+-d——d" — 8.2418553 + .0119631 + 0000694 - 
3 ^9 81 

.0000002=y.2538876=sine of F 1' 40", as was required. 

2. Given the series — , — ,-7:,-7:, — ,&c. to find the term 

50 5152 53 54 



SOMMATIOK OF INFINITE SERIES. 243 

1 

wfcich stands in the middle between the two terms— and 

-i- Ans. -— : 

a3' 105 

3. Given the natural taogents of 88'' 54', 88^ 55', 80" 
56', 88° 57', 88'* 58', 88<> 39', to tind the tangent of 88° 
5Q' n''. Ans. b5.y\ll-ii 



PROBLEM VI. 

Having given a series of equidir'tant terms, a, 6, c, d, e, 
kc. whose tirst differences are small ; to find any interme- 
diate term by interpolation. 

RULE. 

Find the value of the unknown quantity in the equa- 
tion which stands against the given number of terms, in 
the following table, and it will give the t.erm required.* 

1. a~b=0 

2. a -26+0=0 

3. a-36-f3c-c/=0 

4. a — 46-i-6c-4cZ+e=0 

5. a-56410c— 10c^+5e-/=0 

6. a—6b+l5c--20d-\-15e~Gf-\-g—0 

, , n— 1 n— 1 w— 2, , 

a—nb-\-n . — r-—c — n . — - — • — - — »■+• 



Or 



2 2 

n— 1 n— 2 n— 3 



n 



2 



e, &,c. =0. 



EXAMPLES. 



U Given the logarithms of 101, 102, 104, and 105, to 



lind the logarithm of 103. 



* The more terms are given, in any series of this kind, the giore accurate- 
ly y.nW the equation that is to be used approximate towards the "true result, or 
answers required. 



244 SUMMATION OF INFINITE SERIES. 

Here the number of terms are 4. 

And against 4, in the table, we have a-4b-{-Gc — 4d-\-e 

4X[b+d)-(a-\-e) 

=0 ; orc= ^ ;f — ^^^ ^ = value of the unknown 

b ' 

quantity, or term to be found. t, 

a:=2.00432]4 

Where, taking the logs of fc=2. 0086002 

iOl, 102, 104, and J03 ' d=2.0170333 

) 6=2.0211893 

And consequently 

4X(6+d)=i6.K)25340 

a-{-e= 4.0255107 



6)12.0770233 

2.0128372= log. of 103, as 
required. 

2. Given the cube roots of 45, 46, 47, 48, and 49, to 
find the cube root of 60. Ans. 3.684031. 

3. Given the logarithms of 60, 51, 62, 64, 56, and 56, to 
find the logarithm of 63. Ans. 1.7242768695. 

PROMISCUOUS EXAMPLES RELATING TO SERIES. 

1. To find the sum (s) of n terms of the series, I, 2, 
?, 4, 5, 6, &c. 

First, 1+2+3+4+6 &c w=s. 

And»+(n-l)+(n-2)+(«— 3)+(»-4) &c. 

+l=s 

Therefore, by addition, 

(n+l)+(«+l)+(«+l)+(«+l)+(«+l)&c 

+(n+l)=2s. 

And consequently r/(n+l)=2s ; or s= — - — =sum re- 
quired. 

2. To find the sum (s) m n terms of the series, 1, 3, 6, 
7, 9, ll,&ic. 

First, 1+3+6+7+9 &c (2?i— l)=s. 

And (2?i-.l)+(2/i-3)+(2«-6)+ , . . +1=^, 



aUMMATION OF INFINITE SERIES. 245 

Therefore by addition, 

2n+2n+2rt-f 2ra+2w+ &c 2/1=25. 

And consequently 2ftXn=2s ; 

Or s=— — =n3= sum required. 
2 * 

3. Required the sum (s) of n terms of the series, a+ 

First, a + {a-\'d)+{a+2d)+{a-i-3d) &LC + 

{a-\-{n-])d]=s. 

And a-\- (jic/- cZ) 4-a+ {nd~2d) +a+ (nd- 3ti)+a+ 
^nd— 4d) &.C a=s. 

Therefore, by addition, 2a-{-{nd-d)-\-2a+{nd—d)i- 

Sa + (nd~d) &c +2a-\-{nd^d)=2s. 

And consequently (2a-\-nd—d)Xn=-2s ; 

Or s=^{2a-{-nd — d)X-= sum required. 

Or the same may be done in a different manner, as t'ol' 
lows : 

a-\-{a+d) + (a-]-2d) + {a-\-3d)~\-{a-}-id) &c. 
(-{-l + l + I + ] + l&c.)Xa I _ 
(+0+l+24-3+4&c.)XfZ I "^ 
Butn terras of H-l-j-l + l + l Lc.—n. 

And n terms of 0+1+2+3+4 kc /''^^\^'\ 

2 

-.1'. , 7iX(n—])d f, , ,, ,, ?; 

\\ hence s=na+ —- —=\2a -f d (Ji—^jiXc. 

»mf At 

which is the same answer as before. 

4. To find the sum (s) of n terms of the series 1, ar, x^, 
a:^, a;'', &c. 

First, 1 +x+2;2-f-a:3_|_.a:4^ &c. .... a:"-»=s. 

And a;+:r2+.T3+x''+a;5, &,c x"=sx. 

Whence, by subtraction, x"— l=sa;~s, 

x''. 1 

Or s= — = sum required. 

X— ■ 1 ^ 

And, when x is a proper fraction, the sum of the se- 
ries, continued a.d infinitum may be found in the same man 
jier. 

Thus, putting l+x+x^-j-x^+x^+xSj &c.=Si 



246 SU3IMATI0N OF INFINITE SERIES. 

Wr, shall have x-}-x~ -{-x^-t-x'>-\-x^ , &,c.=sx, 
And consequently —l=sx — s; ors — sa;=l, 

Whence s= = sum of an infinite number of terms, 

1 — a; 

as was to be found. 

6. Required the sum (s) of the circulating decimal 

.999999 &c. continued ad vifinitum. 

9 9 9 9 

First, .999999 &c.=— -i -4 r-4— ^ — &c.= 

' 10~iU0~100U 10000 

^^ lT5-^]I.o"^T^+ 1^+ ^^•^='' 

Or, — +— -H — ^4— ^4- &c. =- . 
' 10^100^1000^10000 9 

'^^-^^-^'^+1^+4+1^0-0+^-=^'^ 

And consequently 1=— 5- — q— "5^= ^ » 

y y y 

Whence s=l= sum of the series. 
6. Required the sum (s) of the series a2-|-(a-|-c^)^ + 
(a+2rf)» + (a+3rf)2-}-(a-[-'lci)2,&ic. continued to 71 terms. 

Here 
First, a2 —a'' 

{a-\-dy =c2^2Xlad4-lc?« 
(a-i- 2c/)2 =a2 4-2 X 2af^-i-4rf2 

(0+3^)2 =a2 _|_2 X 3a(^+9rf2 
(a-}-4d)2 =(i2 4-2 X 4ac/-i- 1 Cd2 
&c. &.C. 

Whence 
Sum of n terms of (l + l + l + 14-&c.)a2 
+ . . . ditto of (l + 2+34-4+&c,)2ad 
+ . . . ditto of (l+4 + 94-16+&c.)d 
But n terms of l-j-l-|-l-fl+&c.=n. 

And of 1+2+3+4 &c.-"^"~^^ 



Also of 1+4+9+&C.: 



1 .2 

_ «(«— l)(2n-l) 

' 1.2.3 



SUMMATION OF INFINITE SERIES. 

fe Therefore s=na^ +n{n^ 1 ) arf+- ^^~^^^^"~^^ d^ = 

the whole sum of the series to ?i terms. 

7. Required the sum (s) of the series a^-{-{a-\-d)^-{-(^a 
-f 2(i)^ + (a<+-3(i)3 + (a+4rf)3 &c. continued to n terms. 

First, a 3=: a 3 

(a-f2(/) 3=a3 -1-3 X 2a2 d4-3 X 4(/(i2 _{_g(^3 

(a -i-3rf)3 =«3 4-3 X 3a2 c^-j-3 X Oaci^ -j-STrf^ 
(o+4rf)3=ra3-f-3X4a2(^-t-3xl6arf^ +64(^3 

(a-f- 5(/) 3 =a3 -i-3 X 5a2 d-i-3 X 25ad2 4- 1 25cZ- 
&c. &c. 

Whence 
Sum of n terms of (l-f-l-i-l + l&c.)a3 
4- . . . ditto of (l+2-f34-4&c.)3a«rf 
-f . . . ditto of (l + 44-9+16&c.)3a(/2 
■4- . . . ditto of (l + 8-f27-f64&c.)(i3 

But » terms of l+l-|-l + l + l&c.=:n. 

n(^7i— 1) 



s = 



Ditto 



Ditto 



Ditto 



. . of 1+2+3+4 &LC. 
of 1+4 + 9+16 &c.= 



1 2 

7J(W--I)(27t— 1) 



. of 1 + 8+27+64 &c.= 
Therefore, s=?2a 3 



1.2.3 

n*-2?l3 4-;i2 



(n«— 27l3+„2y3_ 



2X2 
n(n-l)3a3£i ,i(n- l)(2n— l)3ad?2 

2 ' 6 

sum of 71 terms, as was to be found. 



8. Required the sum (s) of 21 terms of the series 1+3 
f-7+ 15+31, &c. 

The terms of this series are evidently equal to 1, (1 + 
2), (1 + 2+4), (1 + 2 + 4 + 8,) &c. or to the successive 
sums of the geometrical series, 1, 2, 4, 8, 16, &c. 

Let, therefore, a=l and r=-2, and we shall have 

a+ar-f-ar--^+Gr3+ar4 &c. =1 + 2+4 + 8 + 16, &c. 

But the successive sums of 1,2, 3, 4, kc, terms of this 
series are, 



248 SUMMATION OF INFINITE SERIES. 



J.— — =(r-l)X- 



Therefore s= — -X 
- r— 1 



r— 1 ^ ' r— 1 

^ ar'^ — a , . a 

2. =fr2-l)X - 

r~-\ - ' r — 1 

„ ar^—a , ,. a 

5 _=(r3-_l)X 

r — I ^ r — 1 

^ ar* — a , . a 
4. _=(r''-l)X 

&c. &c. 

n terms of r-\-r--\-r"-\-r* &.c. 
— Ji, terms of l + l-j-l-f-l &c. 

But 1 + 1-fl+l+l-f 1+1 &C. =71 

And r4-r2+r=4-r*+&c. = (r'^ — 1)X— ^ 

^ ' r— 1 

Whenee s=-!^ --^X - — nX ==2(2"— !)-»= 

r— 1 r— 1 r — 1 ^ 

Tvhole sum required. 

9. It is required to find the sum of n terms of the series. 

1 3 7 15 31 63 „ 

y+2+4+T+61 + 32' ^"' 

Here the terms of this series are the succes&ive sums of 

the geometrical progression — I 1 j 1 &c 

Let, therefore, a=l and r=2, then will 

3^1,1.1. , a a a a n a„ 

-4--+-+-&C. =a-\ j 1 1 1 1 &c. 

1^2 4 8 r~ r^~ r^ r*~ r^~ r'' 

But the Successive sums of 1, 2, 3, 4, &,c. terms of this 

series are, 

(r — ])X1 ^ ^ r— 1 
(r2-l)Xa , 1, a 



(r — 1)X r ^ r^ r— 1 



(r — l)Xr2 ^ r^'' r— 1 

, (r*- l)Xa , K a 

(r -l)Xr' ^ r3^ r-1 

&c. &c- 



SUMMATION OF INFINITE SERIES. u^^ 



a 

;= -X 

r — 1 



Therefore 

n terms of r-{-r-\-r-\-r-\-r &c. 

— n terms ot T"r~+~:rH — ^^'C. 
1 r r- r-^ 



These being the two series derived from the above ex- 



pressions, 



But r-^r-^r-\-r-\-r-\-r kc.=^n.r 

ill 1 , r"— 1 

i r r- 



a 



,3 (r— l)r -» * 

Whence 

r»_l ^_(n— O^n + l _ 

On- 2 



sum requir- 



ed. 

10. Required the sum (s) of the infinite series of the 

reciprocals of the triangular number T+o+g+77;+|T 

kc. 

T 111 1 p 1 ' j: : 

Let — 1 — i 1 &c. ad tnnmium =s. 

1~3^6^10 

Gr 1 1 1 &c =*=s. 

1.1^1.3^2.3^2.5 

That is, (|-^)+(^-i)+(l-I)+(i-l)&e.=|. 

l+i+i+V+i+l&c. 

_1_1_1_1_1_1^^' ~2 

2 



Or, 



1_1_1_1_1^ ' 

"3 4 6 6 7 '^' 



Whence -=- ; or s=2= sum required. 

2 1 

II. And if it be required to find the sum of n terms 01 

1 1 1 1 ■ 1 
le same series, j+g+e+yQ+j^^c. 

11111 1 

Let.=-+2+3+4+^&c.to-. 



250 SUMMATION OF INFINITE SERIES. 

n^. 1 1 1 1 S„ 1 

Then.--=-+-+-+-&c.to-. 

And 2—--] — _= +-+ 4. &c. to——. 
1 n-rl 2 3 4 ' 5 71+1 

vu c 1 * 1 . 1 > ^ , 1 i I 

Jherefoxe -__=^+-+_+-te. to--^. 

Or -4T=i+i+A+:4&c. 10- ' 



n+1 2 ' 6 ' 12 ' 20 * n(n + l)' 

Whence— r—=-^ \---\ &c. to . 

n+1 1^3^6^10 n(n+l) 

Or7+^+^+— +7r&c. to-— --^=— — =sumof nterms 
1 3 6 10 15 72(n-f-lj ?i+l 

of the series, as was required. 

12. Required the sum of the infinite series, A 

'1.2.3^2.3.4 
1_ 1 

■^3.4.5"^4X6^^' 

Let r= ---|---j-_-j--._f._&c. ad infinitum. 

ihen 2r_y=--{-_+_+_&;c. by transposition. 

And ^~f;2'*'£3"^3'4'^45^^' ^7 subtraction. 
Or 1— ^.=- +_+_+_ &c. by transpositioB. 
Whencel=^34-^^+g-A_s.c. by subtraction. 

.2 1-2.3^2.3.4^3.4.5^4.5.6 

1 111 

And--7-2= 1 1 U&c. 

2 1.2.3^2.3.4^3.4.5^ 

But-^2=i ; therefore -4-7;H ^ — h--—-\ — ad 

2 4' 1.2.3^2.3.4^3.4.5^4.5.6 

infinitum =-, which is the sum required. 



SUMMATION OF INFINITE SERIES. 251 

13. And if it were required to find the sum of n terms 
of the same series-^+^+^^+^^&c. 

Andz-.i+ I =l4.L+J_ + l_^l+ 

1 1 

-&c. continued to^-^^-^^^) '^''^'^• 

Therefore I_^-^-_=-.|^+^^+^^ &c. 
to n terms, by subtraction. 

Whencei-^^^^_^/^^^_^^^=^+^^+^-S:c. ion 
terms, by division. 

And consequently— —+——+--— &c. continued ton 

terms =-— ^ , ^ ,^r i o\ = ^""^ required. 
4 2.(n4-l)(»+2) 

14. Required the sum (s) of the series- — 7+0—7^.+ 

— — > &c. continued ad infinitum. 
Let a;=-and s 



2 14-x 

Then-4-=a;(l~x+.T2~a:34-a-*&c.^ 
l-f-x 

And ^=(J+x)X(x— x2+t3-. x^+a;5&;c> 

Whence, by multiplication, 

\-\-x 



Whose sum is =a;4-0+0-fU+0&c. 



252 SUMMATION OF INFINITE SERIES. 



X 

Therefore zsnx, and x — x^ -{• x^ — x* +x5&c. =■; 



1+x 

Or- }-- --f— -&c. =—-7—==-=: sum required. 

2 4^8 16^32 1+i 3 ^ 



2 

12 3 4 
15. Required the sam of the 8eries--f--+r-| — s+^c 

'ontinued ad infinitum. 

Leta:=lands=^^-y^. 

Tben^ — -■^=x-\'2x^-^3x^+4x*-\-5x^kc. 
(1— a;)2 

And z=»(l-rc)2 X(a;+2x2+3x3+4a;44-5.r5&c.) 

Whence, by multiplication, 

a; +2x2 4-3x3+ 4a:* &c. 

1— 2x+x2 



x+2x2 


+3x 


3+4x« 


&c. 


-2x2 


— 4x' 


— 6x3 


&c. 




+x = 


+2x4 


&c. 



Whose sum is =rx+0+0+0+G &c. 
Therefore z=x, 

And x+2x2 +3x3 +4x^+5x5 &c.=- 



(1-X)2- 

infinite series required. 

16, It is required to find the sum (s) of the series -+ 

o 

4 9 16 25 

— +- — I 1 &.C. continued ad infinitum. 

9^27^81^243 -^ 

Let x=-and:^ r:r=s. 

3 (l--a)3 

Then; — ^^-^=x+4x2+9x3 + 16x*+25xs&c. 
(l-x)3 

And ^=(1 x)5 X(r+ix2+9x3 + 16x4&c.)=.T+x2,. 
as will be found by actual multiplication. 



.SUMMATION OF INFINITE bERIES. 253 

Therefore x-\-x-=Zy 

. , a-(l+xO 
And a;4-4j;2+9,r3 + 16a;*&c. +^:^j' 

Or, 

A^l+A^-l^&c. ^iy ii^=-=U= sum required. 
3^9^27^81 (1-i)' 2 - 

a a-\-d , 
17. Required the sum (s) of the series 1 r 

JH — j.^II — : ^, continued ofZ infinitum. 

I '' 

Let x=-, and $: 



r ml 1— x)2 

Then — 1-^=''-+^-'+"-+?'+'^ &c. 

., That is, ^-= 

And z=(l-a;)2 xJa+Ca+rf) a;4.(a-l-2d)x2 + (a+3ri) x^ 

&c.|-(l — x)tt-f-<^a^, 
as will appear by actually multiplying by (1 —xY 

a 
Therefore 2 {l—x)a-\-dx; and consequently \- 



m 



a+d , a-f2d, r {a(r^\)-{-d) . . 

— 1 — -{- — &c.=^ — l-^, ^— ->=.sum of the infinite 

mr mr~ vi ( (r— 1)2 ^ 

series required. 



EXAMPLES FOR PRACTICE. 



1. Required the sura of 100 terms of the series 2, 5, 8, 
>.-, 11, IK &•:. Ans. 16050 

p 2. Required the sum of 50 terms of the series 1+2* 
I .^324.424.52^0, Ans. 42925 

". It is renuirad to find the sum of the scries l4-3x+ 



254 SUMMATION OP INFINITE SERIES. 

G2;2 4-10x3-}- 16x* continued ad infinitum, &.C. when a: ia 
less than 1. 

^"'; (r--i)J 

4. It is required to find the sum of the series l-f-4x 
■\- \Qx^ -\-^lOx'^ -\-Zbx* , &c. continued ad injinitum, when x~ 
is less than J. 

Ans. ,— 

{\-xy 

o. It is'requirfed to find the sum of the infinite series 
^ \- i_+ J- 4- —Sic. Ans. —, or - 

6. Required the sum of 40 terms of the series (1x2) 
■^l-(3X4)-f-(5X6) + (7x8)&c. Ans. 22960 

2jr— 1 

7. Required the sum of 7i terms of the series — - — 

2.r-3 , 2«-6 , 2a,-7 ' . 2x-n, 

i- — ■ — 4- -\ &c. Ans. 7i(— - — ) 

8. Required the sum of the infinite series - - + 

L + — i — + — ? — &c. Ans. -- 

.J.H'.'l. 5 3.4.5.(5 4.6.6.7 18 

9. Required the sum of the series T'^j'^T^'^^o'^sE' 

3 1 

&c. continued c!(Zjn^HJ<«w. Ans. -, or 1^ 

10. It is required to find the sum of n terms of the se 
rieS' l4-8.T-}-27x2 4-64.r-3 + l25.TS &c. 

\+4x-^x- 

Ans. —r- r-- 

(1— a;)* 

• 1 , 2 

1 1 . Required the sum of m terms of the series --f-— -i. 



! ^ZC. 



1 1 inr+r— 1 ) 

Ans. y -X r-l""7 7\~ i 

(r- \ ) r" ^ (r — I ^- ^ 



SUMMATION OF INFINITE SERIES. 255' 

1 1 1 

fc^ 12. Required the sum of the ^^^'^^ ^+r"o+g~^ 



3.12 '^^' ' ' • ' 2»(4+2n) 

_ 3 _ 5/i+3n2 
Ads. 2-yg,s---|-y— -jq^, 

1 11 

. j3. Required the sum of the series ;^g+ .j lo ' "o^Ti; 



• 12.20 ' ■ ' '^3«(4+4») ' 

1 n 

Ans. 2=—-, s= 



12' 12-^-12^ 
6 6 6 

14. Required the sum of the series o~^~^j~[2'^i2~\7 

6 . . 6 



•17:22 ^- • • • ' (6«-3).(o«+2)" 

3 Sn 



Ans. S=-, s: 



'5' £+5» 

1 1 1 

15. Required the sum of the series 3~^""g~Q+9n^0 "^ 

1 1 

3X12"^ ^*^' ' • "3<4T2^' 

1 n n 

^°'' ^"^24' ^=2(3+6'^'" 4(6+6n> 
2 3 4 

16. Required the sum of the series— — ^j'^'f'^'' 

?-.ll"^ • • • -(l+2ft;. (3+^/0 

Ans. S---,s-^2 4(3+4ji* 



* The symbol S) made use of in these, and some of tlie following- series, 
denotes the sum of an infinite number of terms, andS the sum of n terms. 



25Q LOGARITHMS, 

.56 
17. Required the sum of the series 7-77^+;^-^-^+ 

7 8 , 4-fn 






3.4.5 4.*5. 6 ~ n'^l-f/i).(2-t-n) 

3 3 2.1 
Ans. S=-, s: 



* 



2 l4-« 2+n 



Of logarithms. 



Logarithms are a set of nutrsbers that have been com- 
'jjuted and formed into tables, for the purpose of facilitate 
..■,g many difficult arithmetical calculations ; being so con- 
trived, that the addition and subtraction of them answers 
to the multiplication and division of natural numbers with 
which they are made to correspond j. 



* The series here treated of are such as are usually called algebraical; 
Tvhich, of course, embrace only a, small part of the whole doctrine. Those, 
therefore, who may wish for farther iiiformafton on this abstruse but highly 
curious subject are referred to the jypscellanea Annlytica of Demoivre, Ster« 
ling's Mtthod Differ., James Bernoulli de Seri. Injin , Simpson's Math. Dis.-, 
seri., Waring's Medii. Analyt.., Clark's translation of Lorgna's Series, the 
Yirious works of Euler, and Lacroix Traite du Cakul Diff. et Int., ivhere 
lliiy will find nearly all tlie materials that have been hitherto collected re- 
specting- this branch of analysis. 

' f This mode »f computation, which is one of the happiest and jnost use- 
ful discoi'eriesc-'" modern times, is due to Lord Napier, Baron of Merchiston, 
in Scotland, wlio first published a table of these numbers, in the year 1614, 
under the title of Cnnon Mtrificum Logaritkmorum ; which performance 
was eagerly received by the learned throughout Europe, whose efforts were 
innnediatvly directed to the in^provtment and extensions of the new calcu- 
lus, that had so unexpci'tfdiy presented itself. 

Mr. Henry Brijtgs, in particular, who was, at that time, professor of geom- 
cml'm Gicsham College, greatly contribun^d to the advancement of this doc- 
1;itif, not only by the \cvy advantageous alieraf'on which he first introduc-ed 
.into the sys'vjni of these r.umbf^rs, by making 1 <lic logarithm of 10, instead 
of 2 302v>352, as had been dom^ b> Kapler; but also by the publication, in 
1624 and 1633,_pf his two great works, the Ariihmttica Logarithmica and 
ihe Trigonomciria Jirituniea, both of which were formed upon the principle 
above mentioned ; as arc, like'wise, all our common logarithmilic tables, a^ 
present in use. • 



LOGARITHMS. 2o7 

Or, when taken in a similar but. more general sense, 
logarithms may be considered as the exponents of the 
powers to which a given, or invariable, number must be 
raised, in order to produce all the common, or natural 
numbers. Thus, if 

then will the indices x, x', x", kc. of the several powers 
of f.', be tho^^ logarithms of the mimbers y, y', y", &.c. in 
the scale, or system of which a is the base. 

So, that, from either of t!;ese forraulce it appears, that 
the logari'^hm of any number, taken separately, is the in- 
dex of that power of some other number, which when in- 
volved in the usual way, is equal to the given number. 

And since the base a, in the above expressions, can be 
assumed of any value, greater or less than 1, it is pl;un 
that there may be an endless variety of systems of log?- 
ritb.ms, answering to the same natural number. 

it is, likewise, farther evident, from the first of x ese 
equations, that when y^), x will be =0, whatever mr.y 
be the value of a ; and consequently the logarithm of \ \a 
always 0, in every system of logarithms. 

And if a-=l, it is manifest from the same equation tliat 
the base a will be =-y ; which base is, therefore the num- 
ber whose proper logarithm, in the system to which it 
belongs, is 1. 

Also, because ax=y, and a'^'=y\ it follows from the 
mnlti [plication of powers, that wxa^', ova^*^'=]jy; aed 
consequently, by the definition of logarithms, given above, 
a;+a-'=log. yy, or 

log. yy= log. y-\-\og.y'. 
And, for a like reason, if any number of the equa- 
tions ax=y, w^'—y, ax"=y", &c. be multiplied together 
we shall have a'x''^'^x''S(c.—yyy' &c. ; and consequently a;-t' 
■X'-\-x' Sac. =log. yy'y'%.c. : or 

Jog- yy'y" ^c. ; =log. y-{- log. ?/'+ log. y" &c. 



S-^e, for farthe'- details on this part of the subject, the Introduction to my 
Trmiise of Plane and. Spherical Trigonometry, 8vo. 2d Edit. 1313: and for 
the consfruction and us-j of the tables consult 'thos<- of Shcnvin, Hr<ttoit,'Tay- 
lor, Callet, and Borda, where every necessary information of this kind mav 
be readily cblained. 

Z 2 



258 LOGARITHMS. 

From which it is evident, that the logarithm of the pro- 
duct of any number of factors is equal to the sum of the 
logarithais of those factors. 

Hence, if all the fiictors of a given Rumher, in any case 
of this kind, be supposed equal to each other, and the sum 
of them be denoted by m, the preceding property will 
then become 

log. y''=7n log. y. 

From vvhich it appears, that the logarithm of the mih 
povv'er of any number is equal to m times the logarithm of 
that number. 

in hke manner, if the equation ««=?/ be divided by a^'= 
y, we shall have, from the nature of powers, as before, 

— ;, or aT-Wfp- : and by the definition of logarithms, laid 

y 

down, in the first part of this article, a:— a;'=log. - or 

li>g.-,= log. y-log.y'. 

Hence the logarithm of a fraction, or of the quotient 
arising from dividing one number by another, is equal to 
the logarithm of the numerator minus the logarithm of the 
denominator. ^ 

And if each member of the common equation a^=^y be 

m 
raised to the fractional power denoted i^v — , we shall 

" n 

"71 ?Ji 

have, in that case, c «" =^« ; 

And, consequently, by taking the logarithms, as before, 

in , !!' , - 1h in 

-^-a=log. 1,", or log. J/"—- log. y. 

Whcrfe it appears, that the logarithm of a mixed root, 
ov fjower, of any number, is found by multiplying the lo- 
garithm of the given number by the nvimerator of the in- 
dex of that power, and dividing the result by the denomi- 
nator. 

And if the numerator m, of the fractional indes, be in 
this case, taken equal to ], the above formula will then 
T>ecome 



LOGARITHMS. 259 

log. r =- log. y, 
tv 

From which it follows, that the logarithm of the nth 
root of any number is equal to the nth part of the loga- 
rithm of that number. 

Hence, besides the use of logarithms ia abridging the 
operations of multiplication and division, <hey are equally 
applicable to the raising of powers and extracting ofroots ; 
which are performed by simply multiplying the given lo- 
garithm by the index of the power, or dividing it by the 
number denoting the root. 

But although the properties here mentioned are com- 
mon to every system of iogiuithms, it was necessary, for 
practical purposes, to select some one of them from the 
rest, and to adapt the logarithms of all the natural num- 
bers to that particular scale. 

And as JO is the base of our present system of arithme- 
(ic, the same number has accordingly been chosen for the 
base of the logarithmic system, now generally used. 

So that, according to this scale, which is that of the 
common logarithmic tables, the numbers 

. . 10S^10- = , 10-2, 10-', 10% 10', lUS 103, 104, &o. 

Or 

. . —'^, —'—,—-.—, 1, 10, 100, 1000, 10000, &c. 
lOOOU'lOOO lO'J 10 ' ' 

have for their logarithms 
. . . -4, -3, -2, -1, 0, 1, i^, 3, 4, &c. 
Which are evidently a set of numbers in arithmetical 
progression, answering to another set in geometrical pro- 
gression ; as is the case in every system of logarithms. 

And therefore, since the common, or tabular, logarithm 
of any number (n) is the index of that power of 10, which 
when involved, is equal to the givea nuaiber, it is plain., 
from the following equi-tion, 

1 

10c=-r. or 10-^=-, 
'/t 

ihat the logarithms of a!! the intermediate numbers, in the 

above series, may be assigned by aj-proximation, and made 

10 occupy their proper places in the general scale. 



260 



LOGARITHM; 



It is also evident, that the logarithms ot I, JO, 100, 
1000, &.C. being 0, 1,2, 3, &c .respectively, the logarithm 
of any number, falling between 0, and 1, will be and 
some deciaial parts ; that of a number between 10 and 100, 
1 and some decimal parts ; of a number between 100 and 
1000, 2 and some decimal parts ; and so on, for other num- 
bers of this kind. 

And for a similar reason, the logarithms of—, — --, 

— — , &c. or of their equals .1, .01, .001, Lc. in the de- 
scending part of the .scale, being — 1, —2, —3, &c. the 
logarithm of any number, falling between and ] , will be 
— 1, and some positive decimal parts; that of a number 
between .1 and .01, —2 and some positive decimal parts ; 
of a number between .01 and .001, —3, and some positive 
decimal paits ; &.c. 

Hence, likewise, as the multiplying or dividing of any 
number by 10, 100, 1000, &c. is performed bj' barely in- 
creasing or diminishing the integral part of is logarithm 
by 1,2, 3, &c. it is obvious that ail numbers, which con- 
sist of the same figures, whether they be integral, frac- 
tional, or mixed, will have, for the decimal part of their 
logarithms, the same positive quantity. 

So that, in this system, the integral part of any logarithm, 
which is usually called its index, or characteristic, is always 
less by 1 t^an the number of integers which the natural 
Slumber consist of; and for decimals, it is the number 
which denotes the distance of the first significant figure 
from the place of units. 

Thus, according to the logarithmic tables in common 
ase, we have 



JV«/n6ers. 


Logarithms. 


1.36820 


0.1361496 


20,0500 


1.3021144 


335. 2G0 


2.5253817 


.46521 


T.667649P 


-.06154 


2.7891575 


&c. 


&c. 



LOGARITHxMS. 



261 



Where the sign — is put over the index, instead of be- 
fore it, when that part of the logarithm is negative, in or- 
der to distinguish it from the decimal part, which is always 
io be considered as +, or affirmative. 

Also agreeably to what has been before observed, the 
logarithm of 38540 being 4.5859117, the logarithms of 
any other numbers, consisting of the same figures, will be 
as follows : 



JVumbers. 

3854 

385.4 

38.54 

3.854 

.3854 

.0385 J 

.003854 



Logarithms . 
3.5859117 
2.5859117 
1.5859117 
5859117 
T.58o9117 
2 5859117 
3.5859117 



Which logarithm-!, in this case, as well as in all others 
of a similar kind, whether the number contains ciphers 
or not differ only in theii' indices, the decimal, or positive 
part, being the same in them all* 

And as the indices, or integral parts, of the logarithms of 
any numbers whatever, in this system, can always be thus 
readily found from the simple consideration of the rule 
above mentioned, they are generally omitted in the tables, 
being left to be supplied by the operator, as occasion re- 
quires. 

1 ; 

* The great advantages attending tlie common, or Brig-gean system of lo- 
garithms, above all others, arise chiefly from tlie readiness with which we 
can always find tiie characteristic or integral part of any logarithm from the 
bare inspection of t'-ie natural number to which it belongs, and the circum- 
stance,that multiplying or dividing any number by 10, 100, lOOO, &c. only in- 
iluenccs the characteristic of its logarithm, without allecting the decimal part. 
Thus, for instance, if i be made to denote the index or integral part of the lo- 
garitlim of any number n, and d Its decimal part, we shall have log n=» -f- d ; 

N 

log. l(imXN^(i+m)-{-d; log.-— - = (t — m) + d\ where it is plain that 
the decimal part of the logaritlun, in each of these cases, remains the sanie. 



262 LOGARITHMS; 

It may here, also, be fjirther added, that when the loga- 
rithm of a given number in any particular system, is known, 
it will be easy to find the logarithm of the same number in 
any other system, by means, of the following equations, 
a^ =n, and e^'=:n ; or log. n=x, and /. n=x'. 

Where log. denotes the logarithm of n, in the system of 
which a is the base, and /, its logarithm in the system of 
which e is the base. 

For, since aa^=ea^', or 6P=e, and e~i=a, we shall hsve, 
for the base a, -== log. c, or x=x' log. e ; 



X 

and for the base c,- =/. a, or x,'^=-x I. a. 

X 

Whence, by substitution, from the former equations, 

log. n=l. nXlog. e ; or log. ti=l. nX t— , 

l.a 

Where the multiplier log. c, or its equal -■; — expresses 

the constant relation which the logarithms of n hare to 
each other in the systems to which they belong. 

But the only system of these numbers, deserving of no- 
tice, except that above described, is the one that furnishes 
what have been usually called hyperbolic or Nepedan lo- 
garithms, the base e of which is 2.718281828459 . . . 

Hence, in comparing these with the common or tabular 
logarithms, we shall have, b}' putting a in the latter of the 
above formulae= 10, the expression. 

log. n=zl. nX-, — , or I. ?i=log. nXl. 10. 

Where log. in this case, denotes the common tabulai- 
logarithm of the number n, and /. its hyperbolic loga- 
rithm ; the constant factor, or multiplier, y-r- . which is 



LOGARITHIMS. 263 

- being what is usually called the 7rfr.iulus o{ ihe common 
system of logarithms.* 

PROBLEM r. 

To compute the logarithm of any of the natural num 
bers 1, 2, 3, 4, 5, &c. 

RULE. I 

1. Take the geometrical series, 1, 10. 100, 1000, 10000, 
&c. an J apply to it the arithmetical series, 0, 1,2, 3, 4, 

' &c. as logarithms. 

2. Find a geometric mean between 1 and 10, 10 and 
100, or any other two adjacent terms of the series, be- 
twixt which the number proposed lies. 

3. Also, between the mean, thus found, and the near- 
est extreme, tind another geometrical mean, in the same 
manner ; and so on, till you are arrived within the propos- 
ed limit of the number whose logarithm is sought. 

4. Find, likewise, as many arithmetical means between 
the corresponding terms of the other series, 0, 1,2, 3, 

^4, &c. in the same order as you found the geometrical ones, 
9nd the last of these will be the logarithm answering to 
tTie number required. 

, EXAMPLES. 

Let it be required to find the logarithm of T-. 



' * It may here be remarked^ that, although the common logarithms havesu- 
perseded the use of hyperbolic or Neperian logarithms, in ail the ordinary 
operations to which these numbers are generally applied, yet the latter are 
not without some advantages peculiar to themselves; being of frequent oc- 
currence in the application of the Fluxionary Calculus, to many analytical 
and physical problems, where they are required for the finding of certain flu- 
ent;, which could not be so readily determined wit'iout their assistanci; on 
which account great pains have been taken to calculate tables of hyperbolic 
logarithms, to a considerable extent, chiefly for this purrejse. Mr. Barlow, 
in a Collection of Mathematical Tables iatelv put^lished, nas given them for 
the firsi 10000 numbers. 



264 LOGARITHMS. 

Here the proposed number lies between 1 and 10. 

First, then, the log. of 10 is 1, and the log of 1 is 0. 

Therefore ^(lOX l)=v^l0=3. 1622777 is the geo- 
metrical mean ; 

And i(l-{-0)=4^= -5 is the arithmetical meaa ; 
Hence the log. 'of 3.1622777 is .5. 

Secondly the log. of 10 is 1, and the log of 3.1622777 is 
-5. 

Therefore ^(10X3.1622777)=5.6234132 is the geo- 
metrical mean ; 

And i(l4--5)=-75 is the arithmetical mean ; 

Henoft the log. of 5.6234132 is .75. 

Thirdly, the log. of 10 is 1, and the log. of 5.6234132 
is .75; 

Therefore ^(10X5.6234 132) =7.4989422 is the geo- 
metrical mean ; 

And i(l + .75)=.875 is the arithmetical mean ; 

Hence the log. of 7.4989422 is .875. 

Fourthly, the log. of 10 is 1, and the log. of 7.498942^ 
is .875 ; 

Therefore ^(10X7.4989422) =8.6596431 is the geo- 
metrical mean, 

And ^(1-f- -875) = .9375 is the arithmetical mean ; 

Hence the log. of 8.6596431 is .9375. 

Fifthly, the log. of 10 is 1, and the log. of 8.6596431 is 
.9375. 

Therefore y(10X8. 6596431) =9.3057204 is the geo- 
metrical mean, 

And i(l-f. 9375) = . 96875 is the arithmetical mean ; 

Hence the log. of 9.3057204 is .96875. 

Sixthly, the log. of 8.6596431 is .9375, and the log. of 
9.3057204 is .96875 ; 

Therefore ^(8.6596431 X9 3057204) =8.9768713 is 
the geometrical mean, 

And i(.9375+.g6875) =.953125 is the arithmetical 
mean ; 

Hence the log. of 8.9768713 is .953125. 

And, by proceeding in this manner, it will be found, after 
25 extractions, that the logarithm of 8.9999998 is .9542425; 



LOGAKlTHxMS. 265 

which may be taken for the logarithm of 9, as it difiers 
from it so Uttle, that it may be considered as sufficiently 
exact for all practical purpose?. 

And in this manner were the logarithms of all the prime 
i. .numbers at first computed. 

RULE 11. 

When the logarithm of any number (n) is known, the 
logarithm of the next gn^ater number may be readily 
found from the following series, by calculating a sufficient 
number of its terms, and then adding the given logarithm 
to their sum. 

Log.(«+])=log.n4-M' \-^-^i+s{2n+iy +T(iM-T)~ 

J. I 4 i a. ! kcA 

^7(271+0' 9(2n+l)« ' '11(271+1)" > 

Or, 

Log. (n+1) = log. n+ j3-^^ + ^— -+-^_p^ 

5c 7d 9e ^^ i 

'^'l{2n+iy'^ 9(2714-1)2'^ fl(2«-f 1)- ^'S 

WhereA,B,c, &,c represent the terms immediately 
preceding those in which they are first used, and m'= 
twice the modulus=.8685889638 .... * 

EXAMPLES. 

1. Let it be required to find the common logarithm of 
the number 2. 

Here, because n4-l==2, and consequently n=l and 2» 
4-1=3, we shall have 



* It may here be remarked, that the difference belween the logarithms ol" 
any t%vo consecutive iiumbers^^is so much the less a? thi> numbers are greater ; 
and consequently the series whrch comprises the latter part of the above ex- 
pression will in that case converge so much the faster. Thus log. Jt and log". 

(n-f- 1), or its equal log. n -\- log. (1 +- ), will, obviously, difief but littlo 

from each other when n is a large number. 

A a 



26G LOGARITHMS. 

m' .86r5889638 , . 

— = . = .289529654 (a; 

2«+l 3 "^ ^ 

A .289629654 

- . = .010723321 (b) 



3(2n+l)2 3.32 

3b 3X.01U723321 

r-^-== = .000714888 (c) 

5(2w+l)2 6.33 ^ ^ 

5c 5 X. 0007 14888 ^^^^ , , 

■ , , . = -—z = .000056737 (d) 

7(2»-|-l)2 7.32 ^ ' 

7d 7 X. 000056737 
■ 9(2-„+Ty=— 935— = -000004903 (.) 
9e 9 X. 000004903 , , , 

TT(2-„qri7=— inr- = -000000446 (.) 

Uf 1 IX. 00000044 6 
B(S;.+i)i= 13:3^— = -000000042 (o) 

13g 1 3 X. 000000042 
-i5(2»+Ty= U,:^ = -000000004 („) 

Sum of 8 term? . . .301029995 
Add log. of 1 . . , .000000000 



Log. of 2 301029995 

Which Ingaritbm is true to the last figure inclusively. 
2. Let it be required to compute the logarithm of the 
number 3. 

Here, since n+l— 3, and consequently n^=. 2, and 2?; 
+ 1=6, we shall have 

m' 8C858ft964 
2-;:+-,=— 6 ---=-nS7I7793(.) 

A .173717793 

3(2^^4717=— 375^— • =-002316237 (b) 

3b 3 + .0023 16 237 
-5(2mT)^= ___ =.000065590 (c) 

5c 5X 000065590 
^^-H)^- TK. -.000001588 (i>% 



LOGARITHMS. 2p7 

7d 7 X. 000001 588 ^^^^„„„ „, , 

^= =.000000050 (e) 

3;2?i+l)2 9.52 V > 

' 9e 9 X. 000000050 ^„^„^„o , v 

. _= =.000000002 (f) 

il(2rt+l)2 11.52 V ) 

Sum of 6 terms 176091260 

Log. of 2 301039995 

Log. of 3 477121255 

Which logarithm is also correct to the nearest unit \\\ 
the last figure. 

And in the same way we may proceed to find the loga- 
rithm of any prime number. 

Also, because the sum of the logarithms of any two 
numbers gives the logarithm of their product, and the 
difference of the logarithms the logarithm of their quo- 
tient, &c. ; we may readily compute, from the above two 
logarithms, and the logarithm of 10, which is 1, a great 
number of other logarithms, as in the following exam" 
pies : 

3. Because 2X2=4, therefore > .30,029995 

log. 2 ^ 

Mult, by 2 2 

gives log. 4 .GO2059990 

4. Because 2 X 3=6, therefore to > 3010299S"'- 

log. 2 ^ 
add log. 3 .477121255 

gives log. 6 .778151260 

5. Because 23 = 8, therefore log. 2 .301029995 

nmlt. by 3 3 

gives log. 8 .903089985 



268 . LOGARITHMS. 

6. Because 32=9, therefore log. 3 .477121265 

mult, b}' 2 2 



gives log. 9 .954242510 



7. BecauseV=6, therefore from I ^_QQ^^^^^^^ 

take log. 2 
gives log. 5 .698970006 



8. Because 3X4=12, therefore 

to log. 3 



.477121255 



■'»• 



add log. 4 .602069991 



gives log. 12 1.079181246 



And, thus, by computing, according to the general for- 
mula, the logarithms of the next succeeding prime num- 
bers 7, 11, 13, 17, 19, 23, &.C we can iind. by means of 
thesim-^le rules, before laid down for multiplication, divi- 
sion and the raising of powers, as many other logarithms 
as we please, or may speedily examine any logarithm in 
the table. 



MULTIPLJCATION 



BY LOGARITHMS. 

Take out the logarithms of the factors from the table, 
and add them together ; then the natural number answer- 
ing to the sum will be the product required. 

Observing, in the addition, that what is to be carried 
from the decimal part of the logarithms is always affirma- 
tive, and must, therelore, be added to the indices, or inte- 
gral parts, after the manner of positive and negative quan- 
tities in algebra. 



LOGARITHMS. 269 

Which method will be found much more 'convenient, 
to those who possess a shght knowledge of this science, 
^han that of using the arithnietical complements. 

EXAMPLES. 

1. Multiply 37.163 by 4.086, by logarithms, 

37.153 . . . . 1.5699939 
4.086 .... 0.6112984 



Prod. 151.8071 . 2.1812923 



2. Multiply 112.2J6 by 13.958, by logarithms. 
A Of. Logs. 

112.246 .... 2.0491709 
13.958 .... 1.1418232 



Prod. 1563.128 . 3.19.39941' 



3. Multiply 46.7612 by .3275, by logarithms 
J^os. Logs. 

46.7512 .... 1.6697928 

.3275 .... 1.5152113 



Prod. 15.31102 1.1850041 



Here, the -f- 1, that is to be carried from the decimals, 
cancels the — 1, and consequently there remains 1 in the 
'ipper line to be set down. 

4. Multiply .37816 by .04782, by logarithms. 

JVos. Logs. 

.37816 .... r.5776756 
.04782 .... 2.6796096 



Prod. .0180€36 . 2.2572852 



Here the + 1 that is to be carried from the decimals; 

<£> si A^ 



570 LOGARITHMS. 

destroys tlic — 1, in the upper line, as before, and there 
remains the — 2 to be set down. 

5. Multiply 3.768, 2.053, and .007693, together. 

JVbs. Lo^s. 

7.768 .... 0.5761109 
2 063 .... 0.3123889 

.007693 .... 3.8860997 

« 

Prod. .059511 . 2.7745995 

Here the -j- 1 . that is to be carried from the decimals, 
when added to — 3, makes — 2, to be set down. 

6. Multiply 3.586, 2.104C, .8372, and .0294, together 

Nos. Logs. 

3.686 0.554610 

2.1046 323170 

.8372 r.922829 

( .0294 2.468347 

Prod^ .1857618 . . T.268956 

Here the +2, that is to be carried, cancels the — 2, and 
there remains the — 1 to be set down. 

7. Multiply 23.14 by 5. 062 by logarithms. 

An?. 117.1347 

8. Multiply 4.0763 by 9.8432, by logarithms. 

Ans. 40.12383 

9. Multiply 498.256 bv 41.2467, by logarithms. 

Ans. 20561.41 

10. Multiply 4.026747 by .012345, by logarithms. 

Alts. .0497102 

11. Multiply 3.12567, .02868, and .12379, together, by 
logarithms. Ans. .09109706 

12. Multiply 2876.9, 10474, .098762, and .003 1698, by 
Jogarithms. Ans, .0968299 



(271) 



DIVISION BY LOGARITHMS. 



From the logarithm of the dividend, as found in the ta 
hies, subtract the logarithm of the divisor, and the natural 
number answering to the remainder, will be the quotient 
required. 

Observing, if the subtraction cannot be made in the usual 
way, to add, as in the former rule, the 1 that is to be car- 
ried from the decimal part, when it occurs, to the index of 
the logarithm of the divisor, and then this result, with its 
sign changed, to the remaining index, for the kidex of the 
logarithm of the quotient. 

EXAMPLES. 

1. Divide 4768.2 by 36.954, by logarithms, 
Mos. Logs. 

4768.2 .... 3.0783545 
36.964 .... 1.6676616 



Quot. 129.032 . . 2.1106930 



2, Divide 21.754 by 2.4678, by logarithms. 
J^os. Logs. 

21.754 . . . . 1.3375,^91 
2.4678 .... 0.3923100 



Quot. .81518 . . 0.9452291 



3 Divide 4.6257 by .17608, by logarithms. 
J^os Logs. 

4.6257 .... 0.6651725 
.17608 .... 1.2157100 



Quot. 26.274\ . . 1.4194625 



272 DIVISION BY LOGARITHMS. 

Here — !, in the lower index, is changed into •i-l,v.'hich 
is then taken for the index of the result. 
4. Divide .27684 by 6.1576, by logarithms. 

JVos. Logs. 

.27684 .... T.4422i-88 
6.1576 . . . . 0.7124477 



Q,«ot. .0536761 . 2.7297811 



Here the 1 that is to be carried from the decimals, is tak- 
en as — 1 , and then added to — 1 , in the upper index, which 
gives — 2 for the index of the result. 

5. Divide 6.9875 by .075789, by logarithms. 
JVos. Logs. 

6.9875 .... 0.8443218 
.075789 .... 2.8796062 



Quot. ^2,1967 , . 1.9647156 



Here the 1, that is to be carried from the decimals, if 
added to — 2, which makes -—1, and this put down, with 
its sign changed, is -f-l- 

6. Divide .19876 by .0012345, by logarithms. 
JVos. Logs. 

.19876 .... 1.2983290 
.0012345 . . . 3.0914911 



Quot. 161.0051 . 2.2068379 



Here — 3, in the lower index, is changed into -\-3, and 
this added to — 1 , the other index, gives -\-3 — 1 or 2. 

7. Divide 125 by 1728, by logarithms. 

Ans. .0723379 

8, Divide 1728.95 by 1.10678, by logarithms. 

Ans. 1562.144 
2. Divide .1023674 fey 4,96523, by logarithms. 

Ans. 2.061685 



RULE OF THREE BY LOGARITHMS. 273 

I • !0. Divide 19936.7 by .048235, by logarithms. 

Ads. 413739 
il. Divide' .067859 by 1234.39, by logarithms. 

Ans. .000054964« 



THE RULE OF THREE, 



OR PROPORTION, PY LOGARITHMS. 

for any single proportion, add the logarithms of the se- 
cond and third terms together, and subtract the logarithm 
©fthe first t"f om their sum, according to the foregoing rules ; 
then the natural number answering to the result will be 
the fourth term required. 

But if the proportion be compound, add together the 
logarithms of all the terms that are to be multiplied, and 
from the result take the sum of the logarithms of the 
other term?, and the remainder will be the logarithm oi" 
the term sought. 

Or, the same may be performed moEe conveniently thus, 

Find the complement of the logarithm of the first term 
of the proportion, or what it wants of 10, by beginning at 
the left hand, and taking each of its figures fi'om 9, except 
the last significant figure, on the right which must be taken 
from 10 ; then add this result and the logarithms of the 
other two terms together, and the sum, abating 10 in the 
index, will be the logarithm of the fourth term, as before. 

And, if two or more logarithms are to be subtracted, as 
in the latter part of the above rule, add their complements 
and the logarithms, of the terms to be multiplied together, 
and the result, abating as many lO's in the index as there 
are logarithms to be stibtracted, will be the logarithm of 
the term required ; observing, when the index of the lo- 
garithm, whose complement is to be taken, is negative, to 
add it, as if it were affirmative, to 9 ; and then take the 
yest of the figures from 9, as before 



274 RULE OF THREE BY LOGARITHMS. 



EXAMPLES. 



1. Find a fourth proportional to 37.125, 14.768, and 
136.279, by logarithms. 

Log. of 37.125 . . . 1.5696665 



Complement .... 8.4303335 
Log. of 14.768 . . . 1.1693217 
I#>g. of 135.279 . . . 2.1312304 



Ans. 53.81099 . . 1. 7^308856 



2. Find a fourth ppoportionl to .05764, .7186, and 
«34721, by logarithms. 

Log. of .06764 . . . 2.7607240 



Complement .... 11.2392760 

Log. of .7186 . . . r.8664872 

Log. of .34721 . . . T. 5405922 

Ans. 4.328681 . . 0.6363554 



3. Find a third proportional to 12.796 and 3.24718, by 
Jogarithms. 

Log. of 12.796 . . l.-!070742 

Complement .... 8.8929258 

Log. of 3.24718 . . . 0.5115064 

Log. of 3.24718 . . . 0.5115064 

Ans. .8240216 . . 1.9159386 



4. Find the interest of 279/. 5s. for 274 days, at 4-1 
per cent, per annum, by logarithms. 



INVOLUTIOxM BY LOGARITHMS. 275 



Comp. log. of 100 
Corap. log. of 365 
Log. of 279.25 . 
Log. of 274 . . 



Log. of 4.5 



8.0000000 
7.4377071 
2.4459932 
2.4377506 
0.653-^125 



Ads. 9.433296 . . 0.9746634 



5. Find a fourth proportional to 12.678, 14.065, and 
100.979, by logarithms. Ans. 112.0263 

6. Fir.d a fourth proportional to 1.9864, .4678, and 
60.4567, by logarithms. Ans. 11.88262 

7. Find a fourth proportional to .09658, .24958, and 
.008967, by logarithms. Ans. .02317234 

8. Find a mean proportional between .498621 and 
2.9587, and a third proportional to 12.796 and 3.24718 by 
logarithms. Ans. 17.55623 and .8240216, 



INVOLUTION, 



OR THE RAISING OF POWERS BY LOGARITHM*). 

Take out the logarithm of the given number from the 
tables, and multiply it by the index of the proposed pow- 
er ; then the natural number, answering to the result, will 
be the power required. 

Observing, if the index of the logarithm be negative, 
that this part (jf the product will be negative ; but as what 
^s to be carried froai the decimal part will be affirmative, 
the index of the result must be taken accordingly. 



E.XAMPLES. 

t. Find the square of 2.7568, by logarithms. 



276 INVOLUTION BY LOGARITHMS. 

Leg. of 2.7568 . . . 0.4402477 



Square 7.599946 . . 0.8804954 
2. Find the cube of 7.0851, by logarithms 

3 



Log. of 7.0851 . . . 0.8503.'i99 



Cube 355.6475 . . . 2.5510197 
3. Find the fifth power of .87451, by logarithms, 

Log. of .87451 . . . r.9417648 

5 



;Fifth power .5114695 . 1.7088240 



Where 5 times the negative index — 1, being— 5, and +4 

to carry, the index of the power is 1. 

4. Find the 366th power of 1.0045, by logarithms 
Log. 1.0045* . . . 0.0019499 

365 



97495 
116994 
58497 



Power 5.148888 . 0.7117135 



5. Required the square of 6.0598.7, by logarithms. 

Ans. 36.72203 

6. Required the cube of .176546, by logarithn:)S. 

Ans. .005502674 



* This answer 5.143P88 though found strictly according to the general 
rule, is not correct in \.h'^ last four figures 8888; nor can the answers to such 
questions relating' to very hirfi powers be generally found true to 6 places ol 
figures by the tablrs ot Log. commonly used ; if any power above the hun- 
dred thousandtii were r.;quired, not one figure of the answer here given 
could be d( pendod on. The Log. of 1.0045 is 00194994103 true to eleven 
places, which muhiplied by 365 gives .7117285 true to 7 places, and the 
correspondiiig number true to 7 places is 5.149067. See Doctor Adrain'* 
addition of Hut. Math. Voi. I. p. 169. 



EVOLUTION BY LOGARITHMS 2>7 

7. Required the 4th power of .076543, by logarithms. 

Ans. .00003 13269 

8. Required the 5th power of 2.97643 by logainthms. 

Ans. 233.6031 

9. Required the 6th power of 21.0576 by logarithms. 

Ans. 87187340 

10. Required the 7th power of 1.09684, by logarithms. 

Ans. 1.909864 



EVOLUTION, 



OR THE EXTRACTION OF ROOTS, BY LOGARITHMS. 

Take out the logarithm of the given number from the 
table, and divide it by 2, for the square root, 3 fur the cube 
root, &c. and the natural number answering to the result 
will be the root required. 

But if it be a compound root, or ene that consists both 
of a root and a power, multiply the logarithm of the given 
number by the numerator of the index, and divide the pro- 
duct by the denominator, for the logarithm of the root 
sought. 

Observing, in either case.-when the index of the loga^ 
rithm is negative, and cannot be divided without a remain- 
der, to increade it by such a number as will render it ex- 
actly divisible ; and then carry the units borrowed, as so 
many tens, to the lirst figure of the decimal part, and divide 
the whole accordingly. 

EXAMPLES, 

I. Find the square raot of 27.465, by logarithiris. 
Log. of 27.465 . . 2)1.4387796 



Root 6.2407 . . . .7193898 

Bb 



278 EVOLUTION BY LOGARLTHMto 

2. Find the cube root of 35.6415, by logarithms: 
Log. of 35.6415 . . . 3)1.5519560 

Root 3.29093 . . . .5173186 



3. Find the 5th root of 7.0325, by logarithms. 
Log. of 7.0825 . . . 5)0.8501866 

Root 1.479235 ... .1700373 



4. Find the 365th root of 1.045, by logarithms. 
Log. of 1.045 . . 365)0.0019499 

Root 1.000121 . . . 0.0000534 



5. Find the value. of (.001234)^ by logarithms. 

Log. of 001234 . . . 3 0913152 

2 



3)6.1820304 



Ans. .00115047 . . . 2.0608768 



Here, the divisor 3 being contained exactly twice in the 
negative index —6, the index of the quotient, to be put 
down, will be —2. 

3 

Find the value of (.024554)^ by logarithms. 
Log. of .024554 . . . 2.39012^3 



2)5.1703669 



Ans. .00384754 . . , 3.5851834 



Here 2 not being contained exactly in — 6, 1 is added to 
if. 'vVhich gives —3 for the quotient ; and the 1 that is bor 



QUESTIONS IN LOGARITHMS. 279 

rowed being carried to the next figure, makes 11, which, 
divided by 2, cives .58, kc. 

7. Required the square root of 365. 5674, by logarithms. 

Ans. 19.11981 

8. Required the cube root of 2.S87636, by logarithms. 

Ans. 1.4402.65 

9. Required the 4th root of .967845, by logarithms. 

Ans. .9918624 

10. Required the 7th root of .098674, by logarithms. 

Ans. .7183146 

21 ^ 

1 1. Required the value of (— -)^, by 'logarithms. 

.\ns. .146895 

112 5 
•2. Requifed the value of (j^z^y, by logarithms. 

Ans. .1937115 



MISCELLANEOUS EXAMPLES I.N LOGARITHMS. 

2 

1. Required the square root of — — , by logarithms. 

Ans. .12751o3 

2. Required the cube root of . . , - q1 ^y logarithms. 

Ans. .6827842 
3> Required the .07 power of .00563, by logarithm.'. 

Ans. .69588S! 
1. 1 

4. Required the value of >^ T^"^' by logarithms 

Ans. .04279825 

15 7 

.J. Required the value of -^-X. 0123/ — ^ by loga- 
rithms. Aus. .001165713 

6. Required the value of ■''^ - -^ -—- '^ ■ ^ bylo- 

' 7i3/l2iX.l9*/17i ^ 

garithms. Ans. .30.09168638 



280 JVnSCELLANEQUS QUESTIONS. 

1. iieq,„red t„e value of -_-(l>^-^_p, by 
logarithms. Aus. 49.o8712 

MISCELLANEOUS QUESTIOxNS. 



1. A person beir>g asked what o'clock it was, replied 
that it was between eight and nine, and that the hour and 
minute hands were exactly together ; what was tlie time ? 

Ans 8h. 43min. 38fY sec. 

2. A certain number, consisting of two places of fig- 
ures, is equa! to the difference of the squares of it? digits, 
and if 36 be added to it the digits will be inverted ; what 
is the number? Ans. 48 

3. What two numbers are those, whose difference, suns, 
and product, are to each other as the numbers 2, 3, and 6, 
respectively ? Ans. 2 and JO 

4. A person, in a party at cards, betted three shillings 
to two upon every deal, and after twenty deals found he 
had gained live shillings ; how many deals did he win ? 

Ans. 13 

5. A person wishing to enclose a piece of ground with 
palisades, found, if he set them a foot asunder, that he 
should have too few by 150, but if he set them a yard 
asunder he should have too many by 70 ; how many had 
he? Ans. 180 

6. A cistern will be filled by two cocks, a and b, run- 
ning together, in twelve hours, and by the cock a alone 
in twenty hours ; in what time will it be filled by the cock 
B alone ? Ans. 30 hours 

7. If three agents, a, b, c, can produce the effects a, 
h,c, in the times e,/, g, respectively ; in what time would 
they jointly produce the effect d. 

Ans. cZ-r(-+74--) 



o 



IvIISCELLANEOUb QUESTIONS. 281 

£. What number is that, which being severally added 
to 3, 19, and 51, shall make the results in geometrical pro- 
gression ? Ans. 13 

9. It is required to find two geometrical mean propor- 
tionals between 3 and 24 , and four geometrical mean? 
between 3 and 96. 

Ans. 6 and 12 ; and 6, 1^, 24, and 48 

10. It is required to find six numbers in geometrical 
progression such, that their sum shall be 315, and the sum 
of the two extremes 165. 

Ans. 5, 10, 20, 40, 80, and 160 

11. The sum of two numbers is a, and the sum of their 
reciprocals is b ; required the numbers. 

12. After a certain number of men had been employed 
on a piece of work for 24 days, and had half finished it, 
16 men more were set on, by whicJi the remaining half 
was completed in 16 dnys ; bow ai-'itiy mun were employ- 
ed at first ; and what was the whole e> ru-nce, at Is. 6d. a 
day per man ? Ans. 32 the nurr-ber of men ; and the 

wboie expence 1 15/. 45. 

13. It is required to find two numbers such, that if the 
square of the first be added to the second, the sum shall 
be 62, and if the square of the second be added to the 
first, it shall be 176. Ans. 7 and 13 

14. The fore wheel of a carriage makes six revolu- 
tions more than the hind wheel, in going lx!0 yards ; but 
if the circumference of each wheel was in^-reased by 
three feet, it would make only four revolutions more than 
the hind wheel in the same space ; what is the circumfe- 
rence of each wheel ? An^. 12 and 15 feet 

15. It is required to divide a given number a into two 

such parts, x and y, that the sum of rnx and ny shall be 

•equal to some other given number b, 

. ha — n , am — h 

Ans. x~ and s/— 

■m — n in—n 

16. Out of a pipe of wine, containing 84 gallons, 10 
gallons were drawn off, and the vessel replenished with 

B J32 



282 MISCELLANEOUS QUESTIONS. 

10 gallons of water ; after which, 10 gallons of the mix- 
ture were againdcawn off, and then 10 gallons more of wa- 
ter poured in ; and so on for a third and fourth time ; 
which being done, it is required to find how much pure 
wine remained in the vessel supposing the two fluids to 
have been thoroughly mixed each time ? Ans. 48| gallons 

17. A sum of money is to be divided equally among a 
certain number of persons ; now if there had been 3 clai- 
mants less, each would have had 150/. more, and if there 
had been 6 more, each would have had 120/. less ; required 
the number of persons, and the sum divided. 

Ans. 9 persons, sum 2700/. 

18. From each of sixteen pieces of gold, a person Mied 
the worth of half a crown, and then offered them in pay- 
ment for their original value, but the fraud being detected, 
and the pieces weighed, they were found to be worth, in 
the whole, no more than eight guineas ; what was the ori- 
ginal value of each piece ? Ans. 13s. 

19. A composition of tin and copper, containing 100 
cubic inches, w^s found to weigh 505* ounces ; how many 
ounces of each did it contain, supposing the weight of a 
cubic inch of copper to be 6^ ounces, and that of a cubic 
inch of tin 4^ ounces. 

Ans. 420 oz. of copper, and 85 oz. of tin 

20. A privateer running at the rate of 10 miles an hour, 
discovers a vessel 18 miles a head ©f her, making way at 
the rate of 8 miles an hour ; how many miles will the lat- 
ter run before she is overtaken. Ans. 72 miles 

21. in how many different ways is it possible to pay 
iOO/. with seven shilling pieces and dol!ars of 45. 6d. 
each ? Ans. 31 different ways 

22. Given the «um of two numbers = 2, and the sum 
of their ninth powers =32, to find the numbers by a quad- 
ratic equation. Ans. \±Ia^/ {b^3i -S3). 

23. (iiven y^ — ar^=666, and x^+a:?/— 406 ; to find x 
and y. Ans. x=7, and y=9. 

24. The arithmetical mean of two numbers exceeds the 
geome-trical meiin by 13, and the geometrical mean exceeds 
the harmonica! mean by 12 ; what are the numbers ? 

Ans. 234 and 104 



MISCELLANEOUS QUES'J'IONS. 283 

26o Given x^y+y^x=-3, and x^y'^-{-y'^x^=7,{o find the 
values of x and y. 

Ans. a;=i(v'.5+l),2/=KA/5~l) 

26. Given x-\-y-\-z=23, xy-\-xz-^yz=l{jl, and xy2= 
385, to find x, y, and z. Ans. x— 5, y=7, z=ll 

27. To find four numbers, a-, 2/, z, and tt-, having the 
product of every three of them given; viz. xyz=23l, 
xyw=4'20, yzw = l540, and :r2'W'=660. 

Ans. a-=3, y=l, z—l 1, and w—20 

28. Given .r+?/z=384, 2/+a;?=237, and z-fa:j/=192, 
to find the values of x, y, and z. 

Ans. 3 = 10, 2/= 17, and 2'=22 

29. Given x^-{-xy=]08, y^-hyz=^6d, and z^+xz=5iiO, 
to find the values of .-r, y, and z. 

Ans. a,'=9, y=3, and 2=20 

30. Given x^+xy-{-y- = 5 and a:*+2;2«3 4-^/* = l 1, to 

find the values of x and y by a quadratic. 

2 1 2 1 

Ans. x=-^W+-^5,y=jy 10^-^5 

31. Given the equation x* —6x3+1 Sx^ — ]2x=5, to 

3 1 

find the value of x by a quadratic. Ans.-±:-y^29 

32. It is required to find by what part of the population 
a people must increase annually, so that they may be dou- 
ble at the end of every century. 

Ans. By 144fh part nearly 

33. Required the least number of weights, and the 
weight of each, that will weigh any niiniber of pounds 
from one pound to a hundred weight. 

Ans. 1, 3, 9, 27, 81 

.34. It is required to find four whole numbers such, that 

the square of the greatest may be equal to the sum of the 

squares of the other three. Ans. 3, 4, 12, and 13 

35. It is required to find the least number, which being 
divided by 6, 5, 4, 3, and 2, shall leave the remainders 6, 
4,3, 2, and 1, respectively. Ans. 59 

36. Given the cycle of the sun 18, the golden number 
S. and the Roman mdiclion lu, to find the year. 

Ans. 1717 



484 MISCELLANEOUS QUESTIONS. 

37. Given 256.T — 872/=l. to find the least possible values- 
of X and ^^ in whole numbers. Ans. ar=52, and 2/=153 

38. It is required to find two different isosceles triangles 
such, that their perimeters and areas shall be both express- 
ed by the same numbers. 

Ans. Sides of the one 29, 29, 40 ; and of the other 37, 37, 24 

39. It is required to find the sides of three right angled 
triangles, in whole numbers, such, that their areas shall 
be all equal to each other. 

Ans. 58, 40, 42 ; 74, 24, 70 ; 113, 15, 112 
j_ 

40. Given a; 3;= 1.2655, to find a near approximate va- 
lue of X. ' Ans. 3.82013 

41. Given a;ys«=5OO0, and z/a; = 3000, to find the values 
«f X and y. Ans. 3:=4. 691445, and t/=5. 510132 

42. Given a-^+2/y=285, and t/^ — xy = 1 4, to find the 
values of x and y. Ans. a;=4. 016698. and 2/=2.8257l6 

43. To find two whole numbers such, that if unity be 
added to each of them, and also to their halves, the sums, 
in both cases, shall be squares. Ans. 48 and 1680 

44. Required the two least nonquadrate numbers x and 
V such, that x^-\-]^' and x^^y"^ shall be both squares. 

Ans. x=364 and ?/=273 

45. It is required to find two whole numbers such that 
their sum shall be a cube, and their product and quotient 
squares. Ans. 25 and 100, or 100 and 900, &c. 

46. It is required to find three biquadrate numbers such, 
that their sum shall be a square. Ans. 1 2'» , 1 5* , and 20* 

47. It is required to find three numbers in continued 
geometrical progression such, that their three differences 
shall be all squares. Ans. 667, 1008, and 1792 

48. It is required to find three whole numbers such, that 
the sum or difference of any two of them shall be ^square 
numbers. Ans. 434657, 420968, and 1.50568 

49. It is required to find two whole numbers such, that 
theiv sum shall be a square- and the sum of their squares a 
biquadrate. Ans. 4566486027761 and 1061552293520 

60. It is required to find four whole numbers such, that 
the difference of every two of them shall be a square num- 
ber. Atts. 1873432, 2288168, 2399057, and 6560657 



MISCELLANEOUS QUESTIONS. 285 

1 2 
bl. It is required to fiod the sum of the series -+q 

3 4 ... 3 

J 1 +&C. continued to infinity. Ans. - 

' 27^81 •'4 

52. It is required to find the sum of the infinite series 

3 9 , 27 81 , 243 S 
-4 ■ &c. Ans. --. 

4 16^64 266^1024 7 

63. Required the sum of the series 5+6+7-J-8+94- 

kc. continued to n terms. Ans. -^{n-h^p 

54. It is required to find how many figures it would take 
to express the 25th term of the series 2' +22 4-24+28 -{- 
21 ^&c. Ans. .5050446 figures 

65. It is required to find the sum of 100 terms of the 
series (1X2)+(3X4)+(5X6)+(7X8) + (9X 10) &c. 

Ans. 343800 

66. Required the sum of 12+22+32+42 + 52 kc. . . 
. +602 which gives the number of shot in a square pile, 

the side of which is 50. Ans. 42925 

67. Required the sum of 25 terms of the series 35+36 
X2+37X3+38X4 + 39X5 &c. which gives the number 
of shot in a complete oblong pile, consisting of 25 tiers, 
the number of shot in the uppermost row being 36. 

Ans, 16576 



286 



APPENDIX. 



OF THE APPLICATION OF ALGEBRA TO 
GEOMETRY. 



In the preceding part af the present performance, 1 have 
considered Algebra as an independent science, and confined 
myself chiefly to the treating on such of its most useful 
rules and operations as could be brought within a moderate 
compass ; but as the numerous applications, of which it is 
susceptible, eught not to be wholly overlooked, 1 shall here 
show, in compliance with the wishes of many respectable 
teachers, its use in the resolution of geometrical problems ; 
referring the reader to my larger work on this subject, for 
what relates more immediately to the general doctrine of 
curves.* 

For this purpose it may be observed, that when aay 
proposition of the kind here mentioned is required to be 
resolved algebraically, it will be necessary, in the first 
place, to draw a figure that shall represent the several parts, 
or conditions, of the problem under consideration, and to 
regard it as the true one. 

Then, having pi-operly considered the nature of the 



* The learner before he can obtain a conipstent knowled^je of the me- 
thod of application above mentioned, must lirst make himself master of the 
principal propositions of Euclid, or of those contained in my Elements of Ge- 
ometry ; in which work he will find all the essential principles of the science 
comprised within a much shorter compass thaji in the former. 

And in such cases where it may be requisite to esteud this mode of appli- 
cation to trigonometry, mechanics, of any other branch of jnathematics, a 
previous knowledge oi" the nature and principles of these eubjects will" be 
equally necessary. 



APPLICATION OF ALGEBRA, &c. 281t 

«^u29tion, the ligure so formed, must, if necessary, be still 
farther prepared for solution, by producing, or drawing, 
such lines in it as may appear, by their connexion or rela- 
tions to each other, to be most conducive to the end pro- 
posed. 

This being done, let the unknown line, or lines, which 
it is judged will be the easiest to find, together with those 
that are known, be denoted by the common algebraical 
•symbols, or letters ; then, by means of the proper geome- 
trical theorems, make out as many independent equations 
as there are unknown quantities employed ; and the reso- 
lution of ihes.e, in the usual manner, will give the solution 
of the problem. 

But as no s^neral rules can be laid down for drawing 
the lines here mentioned, and selecting the propereet quan- 
tities to substitute for, so as to bring out the most simple 
conclusions, the best means of obtainingexperience in these 
matters w*ll be to try the solution of the same problem in 
different ways ; and then to apply that which succeeds the 
best to other cases of the same kind, when they afterwards 
occur. 

The following directions, however, which are extracted, 
with some alterations, from Newton's Universal Arithmetic, 
and Simpson's Algebra and Select Exercises, will often be 
found of considerable ttse to the learner, by showing him 
how to proceed in many cases of this kind, where he would 
otherwise be left to his own judgment. 

1st. In preparing the figure in the manner above men- 
tioned, by producing or drawing certain lines, let them be 
either parallel or perpendicular to some other lines in it, 
or be so drawn as to form similar triangles ; and, if an an- 
■ gle be given, let the perpendicular be drawn opposite to 
it, and so as to fall, if possible, from one end of a given 
line. 

2d. In selecting the proper quantities to substitute for, 
let those be chosen, whether required or not, that are 
nearest to the known or given parts of the figure, and by 
^eans of which the next adjacent parts, may be obtained 
by addition er subtractien only, without using surds. 



288 APPLICATION OF 

3d. When in any problem, there are two lines, or quaa- 
titles, alike related to other parts of the figure or problem, 
the 'ibest w^y is not to make use of either of them sepa- 
rately but to substitute for their sum, difference, or rect- 
angle, or the sum of their alternate quotients ; or for some 
other iine or lines in the figure, to which they have both 
the same relation. 

4th. When the area, or the perimeter, of a figure is 
given, or such parts of it as have only a remote relation 
to the parts that are to be found it will sometimes be of 
use to assume another figure similar to the proposed one, 
that shall have one of its sides equal to unity, or to some 
other known quantity ; as the other parts of the figure, m 
such c.ises. may tken be determined by the known proper^ 
tions of their like sides, or parts, and thence the resulting 
equation required. 

These being the most general observations that have 
hitherto been collected upon this subject, 1 shall now pro- 
ceed to elucidate them by proper examples ; leaving such 
farther remarks as may arise out of the mode of proceed- 
ing here used, to be applied by the learner, as occasion 
requires to the solutions of the miscellaneous problems 
given at the end of the present article. 

PROBLEM I. 

The base, and the sura of the hypothenuse and perpen- 
dicular of a right angled triangle being given, it is requir- 
ed to determine the triangle. 




Let ABC, right angled at c, be the proposed triangle ; 
and put Bc=6, aod ac—o;. 



ALGEBRA TO GEOMETRY. 



209 



Then, if the sum of ab and ac be represented by s, the 
hypothenuse ab will be expressed by s — x. 

But, by the well known property of right angled trian- 
gles (Euc. I. 47) 

AC2-{-BC3=:ab2, Of 

a;2-f-62=52_2.v.r+a;^. 
Whence, omitting x^ , which is common to both sides of 
the equation, and transposing the other terms, we shall 
have 2sx=s^~- b^, or 

^— 2s- ' • • • • 
which is the value of the perpendicular ac ; where s and 
b may be any numbers whatever, provided s be greater 
than b. 

In like manner, if the base and the difference between 
the hypothenuse and perpendicular be given, we shall 
have, by putting x for the perpendicular and rZ-j-a: for the 
hypothenuse. 

x~+2dx+d''-b--\-x^, or 
b'-—d= 

•zd 
Where the base (6) and the given difference (.d) may 
be any numbers as before, provided b be greater than d. 

PROBLEM ir. 



The difference between the di.igonal of a square and 
one of its sides being given, to determine the square. 




* Tlie edition of Eudid, referred to in this and all the fdlloiving' problems 
is that of Dr. Simson, Loiidwi, 1801 ; which may also be used in tiie georac- 
trital construction of these pi-oblems, should the student be inclined to exer- 
cise his talents upon tliis elegant but more difficult branch of the subject 

C C 



290 



APPLICATION OF 



Let AC be the proposed square, and put the side bc, er 
CB>, =a;. 

Then, if the difference of bd and bc be put =d, the 
hypothenuse bd will be =x-\-d. ""^ 

But since, as in the former problem, Bc^-f cd-, or 3Be^ 
=bd2, we shall have 

2x2=^x'+2dx'\-d2,OT 
x^—2dx=d^. 
Which equation being resolved according to the rule 
laid down for quadratics, in the preceding part of the 
work, gives 

x=d-\-dy/2. 
Which is the value of the side bc, as was required. 

PROBLEM III. 

The diagonal of a rectangle abcd, and the perimeter, 
or sum of all its four sides, being given, to find the sides. 




Let the diagonal ac=</, half the perimeter ab-{-bc=g, 
and the base bc=x ; then will the altitude AB=a— x. 

And since, as in the former problem, ab3-|-ec2=ac', 

we shall have 

a" '-2ax-{-x'' ■^x'^ =d^ , or 

d' -a3 
x*~ax= — - — . 

2 

Which last equation, being resolved, as in the former 
instance, gives 

Where a must be taken greater than d and less than d^5 



ALGEBRA TO GEOMETRY. 



291 



PROBLEM IV. 



The base and perpendicular of any plane triangle abc 
feeing given, to find the side of its inscribed square. 




B F D O- C 



Let EG be the inscribed square ; and put bc=6, ad=/), 
and the side o( the square eh or ef=x. 

Then, because the triangles abc, aeh, are similar, (Euc. 
VI, 4,) we shall have 

AD : Bc y. Ai : EH, or 
/) : 6 :: (p—x) : X. 
Whence, taking the products of the means and extremes, 
<here will arise 

px=:bp-~bx. 
Which by transposition and division, gives 

6+/)" 
Where h and p may be any numbers whatever, either 
whole or fractional. 

PROBLEM V. 

Having the lengths of three perpendiculars, ef, eg, eh, 
drawn from a certain point e, within an equilateral triangle 
ABC, to its three sides, to determine the sides. 




f92 



APPLICATION OF 



Draw the perpendicular ad, and having joined ea, eb, 
and Eg, put KF=a, eg=6, eh—c, and bd (which is Ibc) 

Then, since ab, bc, or ca, are each =2x, we shall have, 
hy Euc. 1, 47, 

ad — ./(AB2-BD2)=^(4a:2-x^)=^3a;2=x^3. 
And because the area of any plane triangle is equal to 
half the rectangle of its base and perpendicular, it follows, 
that 

A ABC=::lBC X AD = xXx^3 = X^ ^3, 

ABKC=.JBcXEF=a:Xa =ax, 

AAKC=iACXEG=a?X6 =bx. 



AAEB = iABXEH = a:Xc = 



ex. 



But the last three triangles bec, aec, aeb, are together, 
equal to the whole triangle abc , whence 
ar ^3^=ax-}-bx-\-cx 

And, consequently, if each side of this equation be di- 
vided by X, we shall have 

a\/3=a+64-c, or 

a-^b-\-c 
X = . 

Which is, therefore, half the length of either of the 
three equal sides of the triangle. 

CoK, Since, from what is above shown, ad is = x^3, 
it follows, that the sum of all the perpendiculars, dr^wn 
from any point in an equilateral triangle to' each of its sides, 
iS equal to the whole per; eudicular of the triangle. 



PROBLEM VI. 



Through a given point p, in a given eircle acbd, to draw 
a cord cd, of a given length. 




ALGEBRA TO GEOMETRY. 293 

€)raw the diameter apb ; and put cd = a, ap = b, PB=e, 
and cp=a; ; then will PD=a— x. 

But, by the property of. the circle (Euc. iii. 35,) cpx 
PD=APXi'B; tvhence 

a;(a— a;)=Ac, or 
x-2 —ax= — be. 
Which equation, being resolved in the usual way gives 

x=ia + ^{ia^''bc); 
Where x has two values, both of which are positive. 

vROBLEftf vrr. 

TJirough a given point p, without a given circle abdc, to 
draw a right line so that the part cv, intercepted by the 
circumference, shall ba of a given length. 




Draw PAB through the centre o ; and put CD=a, PAn=:6, 
^E=c, and pc=a; ; then will pd=.t+"- 

But, by the property of the circle, (Euc. ni, 36, cor. ;■ 
pcXPt)=PA Xpe ; whence 

a-(x-T-o)=6c, or 
x'-' -\-nx=hc. 
Which equation being resolved, as in the former prob- 
lem, gives 

Where one value of x is positivje and the other nega- 
tive.* 



* The two '.ast problems, with a few slight alterations, may be readily em- 
ployed for finding the roots of quadratic equations by construction ; but this, 
as well as the metliods frequently given for consti-uctiug cubic and some of 
the higher orders of equations is a matter of little importance in the presen; 
state of mathematical science; analysis, in these cases, beinjgenerally though. 
a.raore-conunodious instrunaent than geometry 



294 APPLICATION OF 



PROBLEM VIII. 



The base bc, of any plane triangle ABc.^the sum of the 
sides AB, AC, and the line ad, drawn from the vertex to the 
middle of the base, being given, to determine the triangle 




B D C 

Put BD or Dc=o, AV—b, AB-j-Ac^=s, and ab=x; then 

will AC=S — X. 

But, by my Geometry, B. ii, 19, AB2-f ac2=2bd2+^ad2; 
wbence 

a;2 4-(s— x)3 =2a2 4-262 , or 
a;2 _sx = 0= 4-62 — is2 . 

Which last equation, being resolved as in the former 
iBstances, gives 

for the values of the two sides ab and ac of the triangle ; 
taking the sign 4" for one of them, and — for the other and 
observing that a'-\-b'^ must be greater than ^s^. 

rjlOBLEM IX. 

The two sides ab, ac, and the line ad, bisecting the ver- 
tical angle, of any plane triangle, abc, being given to find 
ihe base bc. 

A 




ALGEBRA TO GEOMETRY. 



^96 



Put AB=a, AC— 6, AD=c, and bc=x ; then, by Eac. vi. 
3, we shall have 

AB(a) : Ac(6) :: BD ^ DC. 

And, consequently, by the composition of ratios (Euc. 
y, 18,) 

• > ■» «a; 

a-j-b : a :: x : bd- 



and 
a-\-k : b '.', X : dc! 



a+b' 
bx 



a+& 



But, by Euc. vi, 13, bd xdc-{- ao'^=abXa.c ; where 

fore, also, 

ahx'^ 
— --— --f.c2=a6, or 
{a+by 

abx^—(-a+byx{nb-c^). 

From which last equation we have 

, , ,. ab — c"^ ; 

Which is the value of the base bc, as required. 



PROBLEM X. 



Having given the lengths of two lines ad, be, drawn 
from the acute angles of a right angled triangle abc, to the 
middle of the opposite sides, it is required to determine 
the triangle. 

A 




Put AD=a, BE =&, CD or icB=x, and CEor ^ca=i/ ; then, 
since (Euc. i, 47) cd2+ca==ad2, and ce^+cb^^^be", 
we shall have 

a;3_j_42y2=:a2, 



S96 



APPLICATK)N OF 



Whence, taking the second of these equationsfromfour 

times the first, there will arise 

15^2 =40^ — 1,2^ or 
4a2— fc2 

^=^/— 15- 
And in like manner, taking the first of the same equa- 
tions from four times the second there will arise 
15a;2=4/;2 -a^, or 
_ 462— a2 

Which values of x and y are half the lengths of the base 
and perpendicular of the triangle ; observing that b must 
be less than 2a, and greater than iu. 



PROBLEM XI. 



Having given the ratio of the two sides of a plane tri- 
angle ABC, and the segments of the ba«e, made by a per- 
pendicular fulling from the verticle angle, to determine th^ 



:triangle. 




Put BD=a, Dcc=6, AB=a;, ac=^, and the ratio of the 

sides as m to n. 

Then, since by the question, AB : ac : : m : n, and by 
B. II, 16, of my Elements of Geometry, ab* — Ac2=BD2-~ 
Dc2 , we shall have 

X : y '.'. m : n, and 

a;2 ^y2=„S ^{,2^ 

But, hy the first of these expressions, nx=.my^ or y~ 
"^ ; whence, if this be substituted for y in the second. 



m 



there will arise 



ALGEBRA TO GEOMETRY 



£9? 



n-' 

And consequently, by division and extracting the square 
i'oot, ne shall hare 

a2-i»2 
x=m^-- -, and 

a2-62 
m2 — n2 
wbich are the values of the two sides ab, A€, of the than 
gle, 33 was required. 



PROBLEM XU. 



Given the hypethenuse of a right angled triangle abc, 
and the side of its inscribed square do, to find the other two 
sides of the triangle. 




B E 

Put AB=fe, DE, or DF— s, Ac=a', and cb=?/ ; then, by 
similar triangles, we shall have 

Ac(x) : cb(i/) :: AF(a: — s) : fd(s). 
And, consequently, by multiplying the means and ex- 
tremes, 

xy — sy=sx, or 
xy=s{a:-\-y), ... (1) 
But since, by Euc. I, 47, ac2-{-cb2 =ab2, we shall like- 
wise have 

x^^y^^-^h^ (2) 

Whence, if twice equation (I) be added to equation (2), 
there will arise 

x^ -\-^xy-{-y^ =h^ -\-2s{x-\-y), or 
(x-i-y)2 - 2s(x-\-y)—h'', 



I9B 



APPLICATION OF 



Whicli equation, being resolved after the manner of a 
quadratic, gives 

Hence, if this value be substituted for y inequation (1), 
there veill arise 

X j«-a;±y'(A2+s2)| =8 js±y(/j2^sa)| , or 
x"- {»±v/(/i^+*0ja;=-sjs±^(/t«-f-s2)j . 

And. consequently, by resolving this last equation, we 
shall have 

aad 

Which are the values of the perpendicular ac and base 
BC, as nas required. 



FROBLEH XIII. 



Haring given the perimeter of a right angled triangle 
ABC, and the perpendicular cd, falling from the right angle 
•o the hypothenuse, to determine the triangle. 

A 




B C 

Putp=periraeter, cD=a, ac=x, and Bc=y ; then ab= 

But, by right angled triangles (Euc. i, 47) ac2-{-bc3 = 
4b2 ; whence 

•r, by transposing the terms and dividing by 2 

p{x-\-y)-^p''—^y (0 



ALGEBRA TO GEOMETRY. 299 

And since, by similar triangles, ab : bc :; ac: cd, we 
shall also have, by multiplying the means and extremes, 
abxcd=bcXac, or 

ap—a{x+y)=xy (2) 

Whence, by comparing equation (I) with equation (2) 
there will arise 

{a-{-p)'X{x-^y)=ap+ip^. 
Where 

a+p 

And if these ralues be now substituted for x-\-y and y in 
equation (2), the result, when simplified and reduced, will 
gire 

(a-f-/')a;2 ^p{a'\-\p)x — — \ap^ . 

From which last equation and the value of y, aboye 
found, we shall have 



and 

And, if the sura of these two sides be taken fromp, the 
result will give 

Which expressions are, therefore, respectively equal to 
the values of the three sides of the triangle. 

PROBLEM XIV. 

Given the perpendicular, base, and sum of the sides of 
an obtuse angled plane triangle abc, to determine the tw* 
sides of the triangle. 



300 



APPLICATION OF 




C D 

Let the perpendicular AD=p, the base bc=6, the sum 
©f AE and Ac=s, and their difference =x. 

Then, since half the difference of any two quantities 
added to half their sum gives the greater, and, when sub- 
tracted, the less, we shall have 

AB=A(.s+x), and AC=i(s — ar). 

But, by Euc. 1, 47, cd2=ac2— ad^, or cd—^\\(s — 
a:)2_p2 j . and, by B. ii, 12, AB2=Bc2-f ac3+2bcXcd ; 
whence 

And if each of the sides of this last equation be squar- 
ed, there will arise, by transposition and simplifying the 
result, 

(s2_63)a:2=i2^s3. 62)_ 4/^3^2^ or 

4h2 

-=V(>-r^). 

Whence, by addition and subtraction, we shall have 

s . b .. 4p2 



AB: 



'2+-^^^'-.>. 



-62 






'2 



), and 



2^ ^ s«— 62 
Whichiare the sides of the triangle, as was required. 



rRoBLKM xr. 



It b required to draw a right line bfe from one of the 
angles b of a given square bd. so that the part fe, inter- 
cepted by DK and dc, shall be of a given length. 



ALGEBRA TO GEOMETRY. 



301 



A- 



IH 









C 



Bisect FK in g, and put ab or Bc=rt, Foi or GE=Zf, and 
BG =x ; then will BE=a;+^ and BF=a; — 6, 

But since, by right angled triangles, ae2=be3 ~ ab^, we 
shall have 

And, because the triangles bcf, eab, are similar, 



BF : BC 



BE : AE, or 



Whence, by squaring each side of this equation, and ar- 
ranging the terms in order, there will arise 
re" — 2(a2 +^2 )r2 =6= (-2a2 — 62 ). 

Which equation being resolved after the manner of a 
quadratic, will give 

And, consequently, by adding 6 to, or subtracting it from 
this last expression, we shall have 

BE=-^ jaa+62-j-a^(^i2^4j2)| ^^^ or 

BF=^ ja2_{_62-j-a^(a2+462)| -h. 

Which values, by determining the point e, or f, will 
satisfy the problem. 

Where it may be observed, that the point g lies in the 
circumference of a circle, described from the centre », 
with the radius fg, or half the given line. 



PROBLEM xvr. 



The perimeter of a right angled triangle abc, and the 
raiius of its inscribed circle being given, to determine the 
triangle. 

Dd 



302 



APPLICATION or 

A. 




Let the perimeter of the triangle =/), the radius od, or 
OE, of the inscribed circle =r, ae=», and bd=i/, 

Then, since in the right angled triangles aeo, afo, oe is 
equal to OF, and oa is common, af will also be equal ae, 
or X. 

And, in like naanner, it may be shown, that bf is eqnal to 
BD, or y. 

But, by the question, and Euc. i, "47, we have 
{x-{-7-)+{y-\-r}+{a:+y)=p, and 
(^^^ry.^(y+ry={x+yy. 
Or, by adding the terms of the first, and squaring those oi 
the second, 

r(^x-{-y)^^xy— ^"^ • 
Hence, since, in the first of these equations, 2/=(^j7—r) 
—x, if this value be substituted for y in the second, there 

will arise 

x-a _(|p _ r)x -—r{lp- r) . 

Which 'equation, being resolved in the usual mamefj 
gives 

and 

And, consequently, if r be added to each of these last 
expressions, we shall have 

AC=:^(^Lp^r)±^ \\[ip-ry-r{ip-r)\ , 

and 
BC=i(ip+r)q:y \l{^p~ry-r{ip-r)\ , 
for the values of the perpendicular and base of the trian- 
gle, as was required. 



ALGEBRA TO GEOMETRY. 



303 



VROELEM Xvfi. 

From one of the extremities a, of the diameter of a 
given semicircle ade, to draw a right line ae, so that the 
part DE, intercepted by the circumfereace and iipeipendi- 
cular drawn from the other extremity, sh?.U be of a givea 
Fength. 

C 



/' 



\\ 






^N 



\j 



Let the diameter AB=<i,DE=a, and AE-=r ; and Join e3j>. 
Then, because the angle adb is a right angle, (Euc. m, 
31,) the triangles abe abd, are similar. 

And consequently, by comparing their like sides, we 
shall have 

AE : AB : : AB : AD. or 
X : d y, d : x—a. 
Whence, multiplying the means and extremes of these 
proportionals, there will arise 

* x^ -~ax~d^ , 

Which equation, being resolved after the usual manner, 
gives 

PROBLEM XVtir. 

To describe a circle through two given points a,, b, that ^ 
shall touch a right line cd given in position. * 




C I 



G^H 



SOi APPLICATION OF 

Join AB ; and through o, the assumed centre of the re- 
quired circle, draw fe perpendicular to ab ; which will bi- 
sect it in E (Euc. lu, 3). 

Also, join OB ; and draw eh, og, perpendicular to cd ; 
the latter of VY-hich will fall on the point of contact g 
(Euc. HI. 18). 

Hence, since a, e, b, h, f, are given points, put EB=a, 
EF=6, EH=^c, and zo=-x ; which vvill give OF=h — x. 

Then, because the triangle oeb is right angled at e, we 
ghail have 

OB^ =eo*+eb2 , or 
03= ^(^x^ -{-a- ). 

Butj by similar triangles, fe : eh :; ro : og or oe ; or 
h : c ',; b~x : ob ; whence, also, 

OB — -(A— x). 

And, consequently, if these two values of ob be put 
^.quul to each other, there will arise 

^(.,-24. a )=^-{b~x). 

Or, by squaring each side of this equation, and simpli- 

*ying the result, 

(62_c2):c2 4-2ic2a:=fc2(c2_a2). 

Which last equation, when resolved in the usual man- 
ner, gives 



X- 






for the distance of the centre from the chord ab ; 
where 6 must, evidently, be greater than c, and c greater 
■han a. 

PROBLEM XIX. 

The three lines ao, bo, co, drawn from the angular 
points of a plane triangle abc, to the centre of its inscrib- 
ed circle being given, to find the radius of the circle; and 
the sides of the triangle. 



ALGEBRA TO GEOMETRY. 

A 




B EC 

Let o be the centre of the circle, and, on ao produced, 
let fall the perpendiculars cd ; and draw oe, of, or,, to the 
.points of contact e, f, g. 

Then, because the three angle? of the triangle abc are, 
together, equal to two right angles, (Euc. i, 32) the sum 
of their halves oac-j-oca+obe will be equal to one right 
angle. 

But the sum of the two former of these, oac+oca, is 
equal to the external angle doc ; whence the sum of doc 
+OBE, as also of doc+ocd, is equal to a right angle ; and, 
consequently, oBE"-=ocr>. 

Let therefore, Ao=a5 bo=/^, go=c, and the radius oe, 
-OF OF 0G=a;. 

Then since the triangles boe, cod are similar, bo 

t - 

OE : I CO : OD, or 6 : a; : : c : od ; which gives 

OD=-|^, and i:^=^{e- -^^) ov^^^{h"- ^%-). 

Also, because the triangle aoc is obtuse angled at o, v?^ 

-shall have (Euc. ir, 12.) 

Ac2=Ac2--['Co2-t-2AoXoD ; or 

, 2ac.T. ,6(a3-l-c2)4-2ncr. 
Ac=v^(a^ 4-c^ +-^ ) or ^{-^-^^-J^- ) , 

. But the triangles acd, aof, being likewise similars 

AC : CD : : AO : OF, or 

Ma^-{-c^)-\-2acx. c „ 
-v/(-^— ^-y-^- ) : -^v^Ci^-x^) r, a-.'x. 

Whence, multiplying the means and extremes, and squar^ 
ing the result, there v/ill arise 

6;c2 1&(a2 +c3)-f-2ac.r| =a^c^{h^ -x"^). 
Or, by collecting the terms together, and dividijig oj. 
3the coefficient of the highest power of x, 

s d 2 



X 



APPLICATION, &c. 



From which last equation x may be determined, and 
thence the sides of the triangle.* 



PROBLEM XX. 



Given the three sides ab, bc, cd, of a trapezium aecd. 
ini-cribed in a semicircle, to find the diameter, or remain- 



ing side AD. 




Let AB=a, Bc=6, cd=c, and AD=a; j then, by Euc. vi, 
D, acXbd:=adXbc-)-ab XcD=,^x4-«c. 

But ABD, ACD, being right angles, (Euc. iii, 31,) we 
shall have 

AC==^/(AD3_De3), or ■s/{x^-c^), and 
bd=v(ad2 -AB2),or v'C^^-o'*)- 
Whence, by substituting these two values in the former 
expression, there will arise 

^ (.t2 _c2) X v/ {x^ -a2)=i.r+flc. 
Or, by squaring each side, and reducing the result, 
a;3— (a3+fc2-f-c2)af=2afcc. 
From which last equation the value of jrmay be found, 
as in the last problem. t 



* This, and the following- problem, cannot be constrticted geometricallj, 
or by means only of rigrht lines and a circle, being what the eincients usuallj 
denominated solid problems, from the circumstance of their involving an 
eC(uation of more than two dimensions ; in which cases tbey generally employ- 
ed tlie conic sections, or some of the higher orders of curves. 

f Newton, in his Universal Arithmetic, English edition, 1728, has resolved 
this problem in a variety of difierenl ways, in order lo show, that some me- 
tiiods of proceeding, in cases of tiiis kinl fr^q'icntl y lead to more elej^aut so- 
hitions than otliers ; and that a ready knov/ledge of tkese can enly bt obtaiii- 
ed by practice. 



(307) 

MISCELLANEOUS PROBLEMS. 



PROBLEM 1. 



To find the side of a square, inscribed in a given semi 
circle, whose diameter is d. 

1 



Ans.-d^b 
6 



PROBLEM n. 



Having given the hypothenuse (13) of a right angled 
triangle, and the difference between the other two sides 
(7), to find these sides* Ans. 5 and 12 



PROBLEM iir. 



To find the side of an equilateral triangle, inscribed in 
a circle whose diameter is d j and that of another circum- 
scribed about the same circFfe. 

Ans. \d^3, and d'^o 



PROBLEM IV. 



To find the side of a regular pentagon, inscribed in a 
circle, whose diameter is d. Ans. \d^(\0 — 2^ o) 

PROBLEM V. 

To find the sides of a rectangle, the perimeter of which 
shall be equal to that of a square, whose side is a, and its 
area half that of a square. Ans. a-\-\ay/^ and a — 4ay/2 



PROBLEM VI, 



Having given the side (10) of an equilateral triangle, to 
snd the radii of its inscribed and circumscribing circles, 

Ans. 2.8868 and 5.7736 



* Such of these qufstioni a? ane prcposod in mimbers, should first be re- 
solved gcnerall;-, by means Cf iJi#l>?ual «} mbois and then reduced to the aii- 
s^vc^s above given, by substituting the numeral values of the letters iu the re- 
sults thus obtained. 



MB MISCELLANEOUS PROBLEMS- 



PROBLEM Vri. 

Having given the perimeter (12) of a rhombus, and the 
sum (8) of its two diagonals, to tind the diagonals. 

Ans. 4+ v'2 and4 — v/2 

PROBLEM VIH, 

Required the area of a right angled triangle, whose hy- 
pothenuse is x^^ , and the base and perpendicular a;^" and 
x^. Ans. 1.029085 

PROBLEM IX. 

Having given the two contiguous sides (a, b) of a paral- 
lelogram, and one of its diagonals (cZ), to find the other 
diagonal. Ans. v^ (2a2 4-2^2 —d^) 

PROBLEM X. 

Having given the perpendicular (300) of a plane trian- 
gle, the sum of thetwo sides (1150), and the difference of 
the segments of the base (495), to find the base and the 
«ides. , Ans. 945, 375, and 780 

PROBLEM XI. 

The lengths of three lines drawn from the three angles 
of a plane triangle to the middle of the opposite sides, be- 
ing 18, 24, and 30, respectively :, it is required to find the 
sides. Ans. 20, 28.844, and 34.176 

PROBLEM Xn. 

In a plane triangle, there is given the base (50), the 
-area (798), and the difference of the sides (iO), to find the 
-sides and the perpendicular. 

Ans. 36, 46, and 33.261 

fROBLEM XIll. 

Given the base (194) of a plane triangle, the line that 
bisects the vertical angle (66)^and the diameter (200) of 
the circumscribing circle, to find the other two sides. 

Ans. 8i.365§7 and 157.43865 



MISCELLANEOUS PROBLEMS. 309 

PBOBLEM XIV. 

The lengths of two lines that bisect the acute angles of 
a right angled plane triangle, being 40 and 50 respectively, 
it is required to determine the three sides of the trinngle. 
Ans. 35.80737, 47.40728, and 59.41143 

PROBLEM XV. 

Given the altitude (4), the base (8), and the sum of the 

sides (12), of a plane triangle, to find the sides. 

4 4 

Ans. 6+-yoand6— -y'S 

PROBLEM XTI. 

Having given the base of a plane triangle (15), its area 
(.45), and the ratio of its other two sides as 2 to 3, it is re- 
quired to determine the lengths of these sides. 

Ans. 7.7915 and 11.6872. 

PROBLEM XVn. 

Given the perpendicular (24), the line bisecting the 

base (40), and the line bisecting the vertical angle (25) to 

250 
determine the triangle. Ans. The base — ;-\/7 

From which the 'other two sides may be readily found. 

■ PROBLESI XVIII. 

Given the hypothenuse (10) of a right angled triangle, 
and the difference of two lines drawn from its extremities 
to the centre of the inscribed circle (2), to determine the 
base and perpendicular. Ans. 8.08004 and 5.87447 

PROBLEM XIX. 

Having given the ler)gths (a, b,) of two chords, cutting 
each other at right angles, in a circle, and the distance (c) 
of their point of intersection from the centre, to determine 
the diameter of a circle. 

Ans. v' J8(a3-|-6«)-f2c3i 

PROBLEiM XX. 

Two trees, standing on a horizontal plane, are 120 feet 
asTinder ; the height of the highest of which is 100 feet. 



310 MISCELLANEOUS PROBLEMS. 

and that of the shortest 80 ; whereabouts in the plane must 
a person place himself, so that his distance from the top 
of each tree, and the distance of the tops themselves, shall 
be all equal to each other ? 

Ans. 20^^21 feet from the bottom of the shortest, 
and 40^3 feet from the bottom of the other. 

PROBLEM 2Xr. 

Having given the sides of a trapezium, inscribed in a cir- 
cle, equal to 6, 4, 6, and 3, respectively, to determine the 
diameter of the circle. 

Ans. —v/(l30X 153) or 7.061595 

rROBLEM XXII. 

SupposL-.g the town a to be 30 miles from b, b 25 miles 
from c, and c 20 miles from a ; whereabouts must a house 
be erected that it shall be at an equal distance from each 
®f t^em ? Ans. t6. 11 8556 miles from each 

PROBLEM XXIir. 

Given the area (100) of an equilateral triangle abc, 
whose base bc falls on the diameter, and vertex a in the 
middle of the arc of a semicircle ; required the diameter 
of the semicircle. Ans. 20*/3 

PROBLEM XXIV. 

In a plane triangle, having given the perpendicular (p), 
and the radii (r, r) of its inscribed and circumscribing cir- 
cles, to determine the triangle. 

Ans. The base '^^^~^"-^^"-'"). 

PROBLEM XXV. 

Having given the base of a plane triangle equal to 2a, 
the perpendicular equal to a, and the sum of the cubes of 
it's other two sides equal to three times the cube of the 
base ; to determine the sides. 

Ans. a(2+-^6)anaa(2--^6) 



ERRATA. 



N. B. All the lines in the second column are reckoned from the top. 



PAGES. LINES. 

xi 4 For Simpson's, read Simpson. 
For Addition, read Subtraction. 

4c 4c2 

For — -, read +— r"- 
a a 

For 6a read —6a. 
8 9 10 For i2ax, —2xy, and 6a6, read \^a^x, 
— 2a;3y, and Qa*h. 
For —y, read -^y^. 
For 8b, read 8'j* . 
„ 5x2 5jj2 

tor——, read -—-. 
2 ' 26 

For , read . 

a a2 

For Aa^x, road ^a^x. 

For x—\, read x^ —2x-\-\. 

^^ , a 

For y/—,read */— . 

For Xy/x — a2 , read x^/{x — a^ ) , 

For m/a and 2|2/a6, read A\l/a and 2|. 

F©r 3/(6(1 +^2-fv^4)), read ^(6(1 + 
3/24-3/4)). ' 

For -v/(a+v^— 6), »"ca<i ^/(a- V'-ft)* 
For .^(ax-a:^), r«ad v/(a2-.x2). 
For or, (£e/e or. 
For — (a;3-iy), read -(x«-t/»)- 



12 


1 




20 


14 


7 


16 


8 9 1 


17 


16 


26 


25 


41 


17 


42 


8 


51 


16 


53 


22 


55 


11 


39 


24 


66 


15 


71 


18 


72 


14 


74 


12 


76 


4 


.,». 


32 



ERRATA. 

FACES. LINES. 

80 4 8 23 For ^9, -i^— 5, and Cardaus', read i/ 

9, •— ^v^5, and Cardan's. 
86 12 For ^\0-2^2-\-{2-^5)y.\/5, read4 

103 23 For 29i, read 9, 

114 3 13 /brx2+j/3, read x^-^y^y and /or IS-f, 

read +13. 
135 37 /or alO— a;, read Of, 10-ar, 
155 14 For a+c, read d+c. 
157 1 For 2763, ,.gad 27*2. 



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